The continental tectosphere and Earth s long-wavelength gravity field

Transcription

1 Ž. Lithos The continental tectosphere and Earth s long-wavelength gravity field Steven S. Shapiro 1, Bradford H. Hager ), Thomas H. Jordan Department 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 degree 2 36 non-hydrostatic geoid and free-air gravity anomalies onto several tectonic regionalizations. Because both the regionalizations and the geoid have distinctly red spectra, we do not use conventional statistical analysis, which is based on the assumption of white spectra. Rather, we utilize a Monte Carlo approach that incorporates the spectral properties of these fields. These simulations reveal that the undulations of Earth s geoid correlate with surface tectonics no better than they would were it randomly oriented with respect to the surface. However, our simulations indicate that free-air gravity anomalies correlate with surface tectonics better than almost 98% of our trials in which the free-air gravity anomalies were randomly oriented with respect to Earth s surface. The average geoid anomaly and free-air gravity anomaly over platforms and shields are significant at slightly better than the one-standard-deviation level: y11" 8 m and y4" 3 mgal, respectively. After removing from the geoid estimated contributions associated with Ž. 1 a simple model of the continental crust and oceanic lithosphere, Ž. 2 the lower mantle, Ž. 3 subducted slabs, and Ž. 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 to the gravity field, consistent with their keels having small density contrasts. Using this estimate of the platform and shield signal, and previous estimates of upper-mantle shear-wave travel-time perturbations, we find that the average value of Eln rrelnn within the km depth range is 0.04" A continental tectosphere with an isopycnic Ž equal-density. s structure Ž Eln rrelnn s0. s enforced by compositional variations is consistent with this result at the 2.0s level. Without compositional buoyancy, the continental tectosphere would have an average Eln rrelnnsf0.25, exceeding our estimate by 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 Motivated by seismological evidence Že.g., Sipkin and Jordan, 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-wavelength geoid Ž e.g., Kaula, 1967., Jordan Ž proposed that continents are Ž. 1 characterized by thick Ž; 400 km. thermal boundary layers Ž TBLs. which translate coherently during lateral plate motions, Ž. 2 stabilized against small-scale convective disruption by gradients in density due to compositional variations, and Ž. 3 not observable in the long-wavelength gravity r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved. Ž. PII: S

2 136 ( ) S.S. Shapiro et al.rlithos field. The simple plate cooling model, which enjoys much success in describing the structure of oceanic TBLs, cannot be extended to explain thicker continental TBLs Ž Jordan, Instead, Jordan Ž postulated that the thick continental TBL, continental tectosphere, was formed early in Earth s history by advective thickening and has been stabilized against convective disruption by the compositional buoyancy provided by a depletion of basaltic constituents. The isopycnic Ž equal-density. hypothesis Ž Jordan, predicts that the compositional and thermal effects on density cancel at every depth between the base of the mechanical boundary layer and the base of the TBL. Such a structure would be neutrally buoyant with respect to neighboring oceanic mantle, and would not be visible in the long-wavelength gravity field. There has been much discussion during the past 2 decades about the relations among the Earth s longwavelength gravity field, surface tectonics, and mantle convection. For example, there is an obvious association of long-wavelength geoid highs with subduction zones ŽKaula, 1972; Chase, 1979; Crough and Jurdy, 1980; Hager, and with the distribution of hotspots ŽChase, 1979; Crough and Jurdy, 1980; Richards and Hager, Most of the power in the longest wavelength geoid can be explained in terms of lower-mantle structure imaged by seismic tomography Že.g., Hager et al., 1985; Hager and Clayton, 1989; Forte et al., 1993a.. This lower mantle seismic structure has been linked to tectonic processes, in particular, to the history of subduction Ž e.g., Richards and Engebretson, Although there is general agreement among geodynamicists that most of the geoid can be explained in terms of features such as subducted slabs and lower mantle structure, there is significant quantitative disagreement among the predictions of various models Že.g., Panasyuk, Thus, it is not possible to estimate with high confidence the residual geoid not explained by lower mantle structure. The contribution to the geoid of upper-mantle structures, including variations in the thickness of the crust and lithosphere, is a question whose answer is still disputed. Assuming that plates approach an asymptotic thickness of approximately 120 km after cooling about 80 My, the geoid would be expected to be higher by roughly 10 m over continents and over midoceanic ridges than over old ocean basins due to the density dipole associated with isostatic compensation ŽHaxby and Turcotte, 1978; Parsons and Richter, 1980; Hager, At intermediate to short wavelengths, the expected changes in the geoid over these features are observed Že.g., Haxby and Turcotte, 1978; Doin et al., 1996., but the isolation of the geoid signatures of these features at long wavelengths is problematic. Using broad spatial averages over selected areas, Turcotte and McAdoo Ž concluded that there is no systematic difference in the geoid signal between oceanic and continental regions. But, Souriau and Souriau Ž demonstrated that there is a significant correlation between the geoid Ž spherical harmonic degrees l s and the tectonic regionalization of Okal Ž From degree-by-degree correlations Ž l s 2 20., Richards and Hager Ž observed a weak association between geoid lows and shields. On the other hand, Forte et al. Ž reported that the degree 2 8 geoid correlates significantly Ž 99% confidence. with an ocean continent function. Were there a significant ocean continent signal, the continental tectosphere might have a substantial density anomaly associated with it, and might therefore be expected to play an active role in the largescale structure of mantle convection. For example, Forte et al. Ž 1993b. and Pari and Peltier Ž 1996., in their preferred models, assumed linear relationships between seismic velocity anomalies and density anomalies. They proposed dynamic models of the long-wavelength geoid in which the high velocity roots beneath continents are cold, dense downwellings in the convecting mantle. Such downwellings would depress the surface of continents dynamically by about 2 km Ž Forte et al., 1993b.. The lack of significant temporal variation in continental freeboard over geologic time would require that these convecting downwellings be extremely long-lived and translate coherently with the continents Že.g., Gurnis, On the other hand, Hager and Richards Ž and Forte et al. Ž 1993b. found the best fits of their dynamic models to the geoid by assuming an unusually small global proportionality between seismic velocity anomalies and density anomalies in the upper mantle. Forte et al. Ž showed that they could improve their fit to the geoid if they allowed subcontinental regions to have a different proportion-

3 ( ) S.S. Shapiro et al.rlithos Table 1 Ž. GTR1 Jordan, 1981 Region Definition Fractional area Ž %. Oceans 61 A Young oceans Ž 0 25 My. 13 B Intermediate-age oceans 35 C Ž My. Old oceans Ž )100 My. 13 Continents 39 Q Phanerozoic orogenic zones 22 P Phanerozoic platforms 10 S Precambrian shields 7 and platforms ality constant between velocity and density anomalies beneath continents than beneath oceans. To quantify the association of surface tectonics and Earth s gravity field, we investigate the significance of the association between the six-region global tectonic regionalization GTR1 Ž Jordan, ŽTable 1, Fig. 1. and the geoid, EGM96 ŽLemoine et al., 1996., referred to the hydrostatic figure of Earth Ž Nakiboglu, Ž Fig. 2a.. Although we use GTR1 Žand coarser regionalizations created by combining some of these regions. for the bulk of this study, we also compare our results with those obtained using the tectonic regionalizations of Mauk Ž and Okal Ž 1977., as well as the ocean-continent function. Because the geoid spectrum is red, with the rootmean-square Ž rms. value of a coefficient of degree l decreasing roughly as l y2, and because the longest wavelengths are likely dominated by the effects of density contrasts in the lower mantle ŽHager et al., 1985., we also investigate the relationship between GTR1 and free-air gravity anomalies. The gravity field at spherical harmonic degree l is proportional to l y1, so the gravity anomalies are expected to have correspondingly smaller long-wavelength variations than the geoid does. We calculate regional averages of the geoid and the gravity field and estimate their uncertainties. Further, we try to refine the estimate of the contribution 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.

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5 ( ) S.S. Shapiro et al.rlithos subtracting other contributions from the geoid estimates. By combining the upper-mantle shear-wave travel-time anomalies associated with platforms and shields Ž Shapiro, and the results from this study, we estimate, with uncertainties, the average of Eln rrelnns within the depth range km, and compare our estimate with the isopycnic hypothesis Ž. of Jordan Tectonic regionalization and inversion GTR1 and regionalizations published by Okal Ž Ž Table 2. and Mauk Ž Ž Table 3. contain six, seven, and 20 regions, respectively. Both GTR1 and the regionalization of Mauk Ž are defined on a grid of 58=58 cells, whereas the model of Okal Ž is defined using 158 = 158 and 108 = 158 cells. The regionalization of Mauk Ž allows for as many as 10 regions to be represented in a given cell, while the other regionalizations are defined with only one region per cell. In GTR1, the three oceanic regions Ž including marginal basins. are defined by equal increments in the square root of crustal age: 0 25 My Ž A., My Ž B., and )100 My Ž C. and the continental regions are classified by their generalized tectonic behavior during the Phanerozoic: Phanerozoic orogenic zones Ž Q., Phanerozoic platforms Ž P., and Precambrian shields and platforms Ž S.. Like GTR1, the oceanic regions of Mauk Ž are based largely on crustal age. However, the continental regions of Mauk Ž are classified by age rather than by their tectonic behavior. The more complex parameterization associated with the regionalization of Mauk Ž does not offer us any significant advantage over GTR1; as we show through representative projections, the platform and shield signatures from the regionalization of Mauk Ž and from GTR1 are consistent with each other and only significant at slightly better than the one-standard-deviation level. The regionalization of Table 2 Ž. Okal 1977 Region Definition Fractional area Ž %. D Ocean Ž 0 30 My C Ocean Ž My B Ocean Ž My A Ocean Ž )135 My. 2.5 T Trenches and marginal seas 10.9 M Phanerozoic mountains 11.6 S Shields 20.4 Okal Ž is limited in the accuracy of its designation of regions. For example, Okal Ž labels the entire continent of Antarctica a shield, whereas a significant fraction Ž f 1r3. is orogenic in nature. Okal Ž also classifies some islands Že.g., Iceland and Great Britain. as shields. Misidentifications such as these might have a significant effect on results from associated data projections. In general, a tectonic regionalization containing N distinct regions can be described by N functions, R Ž ns1, N. n, each having unit value over its region and zero elsewhere. By combining regions, we can construct other, coarser regionalizations. For example, by consolidating young oceans Ž A., intermediate-age oceans Ž B., and old oceans Ž C. of GTR1, into one region, and Q, P, and S, into another region, we can create a two-component Ž ocean continent. tectonic regionalization Ž ABC, QPS.. For much of this analysis, we combine regions P and S into one region Ž PS.. For any such regionalization, we expand each R n in spherical harmonics, omitting degrees zero and one from our analysis because geoid anomalies are referred to the center of mass and any rearrangement of mass from internal forces cannot change an object s center of mass. With coefficient R lm n represent- ing the Ž l, m. harmonic of region n, and coefficient lm d representing the Ž l, m. harmonic of the observed Fig. 2. Ž. a Geoid, ls2 36 Ž EGM96; Lemoine et al., 1996., referred to the hydrostatic figure of Earth Ž Nakiboglu, 1982.; Ž. b Projection of Ž. a onto Ž A, B, C, Q, PS., and Ž. c Residual: Ž a b.. All plots are displayed using a Hammer equal-area projection with coastlines drawn in white. Negative contour lines are dashed and the zero contour line is thick. The contour interval is 10 m.

6 140 ( ) S.S. Shapiro et al.rlithos Table 3 Ž. Mauk 1977 Region Definition Fractional area Ž %. Oceans Anomaly 0 5 Ž 0 10 My Anomaly 5 6 Ž My Anomaly 6 13 Ž My Anomaly Ž My Late Cretaceous sea floor 21.1 Ž My. 6 Early Cretaceous sea floor 5.4 Ž My. 7 Sea floor older than 140 My 3.5 Continents Island arcs Shelf sediments Intermontane basin fill Mesozoic volcanics Cenozoic volcanics Cenozoic folding Mesozoic orogeny Post-Precambrian undeformed Late Paleozoic orogeny Early Paleozoic orogeny Precambrian undeformed Proterozoic shield Archaean shield 2.0 Ž or model. geoid or gravity field, we use a leastsquares approach to solve: R lm g sd lm Ž 1. n n Ž summation convention implied here and below. for the regional averages, g n. We include the additional constraint: A g s0 Ž 2. n n where An represents the surface area spanned by region n. This constraint ensures that gn have a zero Ž weighted. average, as, by definition, do the geoidheight Ž and free-air gravity. anomalies. The weighted-least-squares solution can be written: T y1 T gs R WR R Wd 3 w x Ž. where the values R lm n and An are the elements of the matrix R, W is a weight matrix constructed from the covariance matrix associated with d, gn are the elements of the vector g, and d lm and zero constitute the vector d. We next consider the effect of errors in d on our analysis. Although we have available the covariance matrix for EGM96, this weight matrix is not the appropriate one for our analysis. As discussed previously, most of the power in the long wavelength parts of the geoid is the result not of surface tectonics, but of deep internal processes. Unfortunately, the contribution of these deep processes cannot be determined to anywhere near the accuracy of the observed gravity field, so the covariance matrix will be swamped by the contributions of the errors due to neglecting important dynamic processes. Quantitative estimation of the errors associated with estimates of the contributions of these deep processes has rarely been attempted ŽPanasyuk Ž 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 geoid increases as l 2 Žor as l for the gravity anomalies.. This behavior is qualitatively consistent with the result that dynamic models of the geoid do better at fitting the longest wavelength components and progressively worse at fitting shorter wavelength components, for example, because the effects of lateral variations in viscosity become more important at shorter wavelengths Že.g., Richards and Hager, The sole exception to the identity weight matrix is our application of a large weight, 1000, to the surface-area constraint. Results from our inversions are insensitive to the value of this weight, so long as it is not less than ten times the weight associated with the data Ž in our case unity. nor so Ž 6 large ) 10 times the data weight. that the inversion becomes numerically unstable. 3. Statistical analysis procedure Because neither the geoid nor the regionalization have white spectra, we do not use common statistical estimates of uncertainties. In fact, their spectra are quite red, implying that uncertainties in parameter estimates based on the assumption of white spectra will be substantially smaller than the actual uncertainties. Through the use of Monte Carlo techniques, we incorporate the spectral properties of these fields in our estimates of parameter uncertainties. For each

7 ( ) S.S. Shapiro et al.rlithos Ž. of 10,000 trials, we 1 randomly select an Euler angle triple from a parent distribution in which all orientations are equally probable and then, in accord with the selected triple, rigidly rotate the sphere on Ž lm lm lm which the data residuals d s d y R g. res n n are de- fined, with respect to the sphere on which the surface Fig. 3. Non-hydrostatic geoid Ž EGM96, ls2 36.: Histograms of parameter values Ž. a g, Ž b. g, Ž c. g, Ž. d g, Ž. A B C Q e g PS obtained from projections onto the tectonic sphere of the correlated data combined with 10,000 random orientations of the data residual sphere, characterized by d lm Ž see text.. Gaussian distributions, determined by the standard deviation, mean, and area of each histogram, are Ž. lm superposed. f Histogram of variance reduction resulting from 10,000 random rotations of the data sphere, characterized by d, with respect to the tectonic sphere. The shaded and unshaded arrows indicate the variance reductions associated with the actual orientation and the maximum variance reduction, respectively.

8 142 ( ) S.S. Shapiro et al.rlithos tectonics are defined Ž tectonic sphere., Ž 2. com lm Ž bine the rotated data residuals d res ; denotes rotated. with the correlated data to produce pseudo data, Ž d lm s d lm lm. Ž. res q R n g n, and 3 project d lm onto Ž A, B, C, Q, PS.. The resulting histograms of parameter values Ž e.g., g, g, g,.... approximate A B C Fig. 4. Free-air gravity Ž ls2 36.: Histograms of parameter values Ž a. g, Ž b. g, Ž c. g, Ž d. g, Ž e. A B C Q g PS obtained from projections onto the tectonic sphere of the correlated data combined with 10,000 random orientations of the data residual sphere, characterized by d lm Žsee text.. Gaussian distributions, determined by the standard deviation, mean, and area of each histogram, are superposed. Ž. f Histogram of variance reduction resulting from 10,000 random rotations of the data sphere, characterized by d lm, with respect to the tectonic sphere. The shaded and unshaded arrows indicate the variance reductions associated with the actual orientation and the maximum variance reduction, respectively.

9 ( ) S.S. Shapiro et al.rlithos Gaussian distributions and, because the correlated signal is added to the rotated data residual before projecting the composite, the resulting histograms of parameter values are centered approximately on the parameter values corresponding to the actual orientation of the data sphere with respect to the tectonic sphere Ž Figs. 3 and 4.. We take these latter parameter values as our parameter estimates and the standard deviations of these approximately Gaussian distributions as the parameter uncertainties. Alternatively, we could assign random Ž white noise. values to each coefficient describing the data-residual sphere while constraining its power spectrum to be unchanged through a degree-by-degree scaling. Histograms resulting from this approach yield very similar distributions and virtually the same values for the parameter estimates and their standard errors Ž Shapiro, If one relaxes the constraint by requiring only that the total power remains unchanged, then the resulting histogram distributions are narrower than the corresponding ones for which the spectra were scaled degree-by-degree. These smaller values for the standard errors in the parameter estimates likely coincide Ž Shapiro, in the limit of large numbers of trials with those determined from the elements of the variance vector 2 w T x z'x diag R WR, where x, is the Žpost- post. 2 post fit x per degree of freedom. As a criterion for the success of the model in fitting the data, we use the percent fractional difference in the prefit and postfit x 2. This percent variance reduction associated with each projection, i.e., w Ž 2 inversion, is thus defined by 100 1y x rx 2.x post pre. From the results of the random rotations of the data sphere with respect to the tectonic sphere, we estimate significance levels in the variance reduction associated with each projection. Specifically, we associate the fraction of trials that yield lower variance reductions than the actual orientation with the confidence level of the variance reduction. 4. Projections Table 4 shows the regional averages and their corresponding statistical standard errors obtained by separately projecting the geoid and the free-air gravity anomalies onto Ž A, B, C, Q, PS.. Fig. 3a efig. 4a e graphically display the 10,000 parameter estimates obtained from the Monte Carlo simulations that lead to the uncertainties given in Table 4. With the geoid, only regions Ž C. and Ž PS. have averages which are larger than their standard errors. However, the significance of these averages is only slightly above the one-standard-deviation level. For example, with 95% Ž 2 s. confidence, the geoid signature associated with platforms and shields is in the range y27 to q6 m, a rather broad range which does not even significantly constrain the sign of this signal. The projection of the geoid onto Ž A, B, C, Q, PS. is shown in Fig. 2b and further demonstrates that very little of the long-wavelength non-hydrostatic geoid can be explained simply in terms of surface tectonics. The magnitude of the geoid signal that is uncorrelated with Ž A, B, C, Q, PS. Ž Fig. 2c. is essentially the same as that of the geoid anomalies themselves, given by EGM96. Using the free-air gravity yields a somewhat different result: four regions have averages larger than their standard errors Ž Fig. 4a e.. The significance of three of these averages is at or below the 1.5s level and the significance of the fourth, g, is at the 2.5s level Ž Table 4. Q. With 95% confidence, the free-air gravity signature associated with platforms and shields is in the range y10 to q1.4 mgal. Like with the geoid, this range is rather large and does not significantly constrain the sign of this signal. However, unlike the geoid projection, which explains less of the variance than about two-thirds of the random orientations of the data sphere Ž Fig. 3f., the free-air gravity projection explains more of the variance than about 98% of projections corresponding with random orientations Table 4 EGM96 Ž ls2 36.: Regional averages and statistical standard errors from projections of the geoid and of perturbations to the free-air gravity onto Ž A, B, C, Q, PS. ga gb gc gq gps Geoid Ž m. 0" " " 16 y4.8" 11 y10.5" 8.3 Gravity Ž mgal. 4.0" 3.4 y1.7" 1.7 y4.8" " 2.5 y4.3" 2.9

10 144 ( ) S.S. Shapiro et al.rlithos of the data sphere Ž Fig. 4f.. However, this reduction in variance is only about 6% and does not produce an impressive fit. Interestingly, Monte Carlo simulations using degrees 2 12 yield confidence levels of less than 30%, suggesting that the association between free-air gravity anomalies and surface tectonics is stronger in the higher frequencies. Using the regionalization of Mauk Ž Žthe full 20-region tectonic sphere as well as some representative groupings of these regions. leads to results similar to those obtained from GTR1. In no case do we find a significant signal that can be linked with the continental tectosphere. Combining the regions of Mauk Ž into three groups based on crustal age Žregions w1 7 x, w8 14, x, w15, x., yields regional averages which are roughly the same magnitude as their corresponding uncertainties ŽTa- ble 5. and a variance reduction of about 6%. Another continental grouping Žw1 7 x, w8 10, x, w11, 14 Table 5 EGM96 Ž ls2 36.: Regional averages and statistical standard errors from projections of the geoid onto several regionalizations based on Mauk Ž Group 1: Žw1 7 x, w8 14, x, w15, x. ; Group 2: Žw1 7 x, w8 10, x, w11, x, w18 20 x. ; Group 3: 20 separate regions Region Group 1 Group 2 Group 3 g Ž m. g Ž m. g Ž m. Oceans 1 9"6 9"6 y13"14 2 9"6 9"6 10"14 3 9"6 9"6 9"8 4 9"6 9"6 5"7 5 9"6 9"6 4"10 6 9"6 9"6 15"14 7 9"6 9"6 40"25 Continents 8 y22"16 y9"15 127"46 9 y22"16 y9"15 y41"19 10 y22"16 y9"15 62"44 11 y22"16 y19"14 y10"57 12 y22"16 y9"15 27"41 13 y22"16 y9"15 6"37 14 y22"16 y19"14 y34"22 15 y7"9 y19"14 4"14 16 y22"16 y19"14 y61"33 17 y22"16 y19"14 y60"26 18 y7"9 y12"13 37"28 19 y7"9 y12"13 y27"15 20 y7"9 y12"13 y23"23 17 x, w18 20x. based instead on a combination of age and tectonic behavior, yields similar Ž insignificant. results Ž Table 5., and even produces a slightly smaller variance reduction than the previous model, which was based on one fewer parameter. On the other hand, when one uses the full 20-region tectonic sphere, the variance reduction associated with the projection of the data sphere is about 20%. This result by itself is not particularly surprising since one would expect the variance reduction to increase with the number of model parameters. However, using this regionalization, less than 10% of our Monte Carlo simulations result in a greater reduction in variance. While this result does not allow us to reject a strong association between the geoid and the surface tectonics defined by Mauk Ž 1977., the large relative uncertainties Ž and even differences in sign. associated with old continents Ž Table 5. suggest that this association is indeed weak. In addition, there are only three regions Ž 8, 9, and 17. that have average values that differ from zero by more than 2s. Although there is substantial uncertainty in the predictions of models of the contribution of other processes to the long-wavelength geoid, perhaps we could better isolate the tectosphere s contribution by subtracting from the observed Ž non-hydrostatic. geoid the effects of previously modeled components: Ž. 1 a simplified representation of the upper 120 km based on the oceanic plate cooling model and a uniform 35-km-thick continental crust Ž l s ŽHager, 1983.; Ž 2. the lower mantle Ž l s 2 4. ŽHager and Clayton, 1989.; Ž 3. slabs Ž l s 2 9. ŽHager and Clayton, 1989.; and Ž 4. remnant glacial isostatic disequilibrium Ž l s Ž Simons and Hager, Separately projecting each of these four contributions to the model geoid onto Ž A, B, C, Q, PS. yields the results given in Tables 6 and 7. Our resulting model Ž residual. geoid, TECT-1 Ž Fig. 5., provides an estimate of the contributions to the geoid of the upper mantle structure below 120 km depth, excluding subducted slabs. For TECT-1, g f-8"4 m Ž PS Table 6.. The projections of TECT-1 separately onto ŽA, B, C, Q, PS., Ž ABC, QPS., and Ž ABCQ, PS. lead to reductions in variance that are listed in Table 7. From the percent of random trials that yield smaller variance reduction than that of the actual orientation Ž confidence level., it is clear that the geoid signal

11 ( ) S.S. Shapiro et al.rlithos Table 6 Regional averages and statistical standard errors from projections onto Ž 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, onto Ž ABC, QPS. and Ž ABCQ, PS. Ž. Ž. Ž. Ž. Ž. Geoid contributors g m g m g m g m g m A B C Q PS Upper 120 km 4.3" 1.0 y3.1" 0.5 y6.1" " " 0.8 Lower Mantle y5" 32 20" 19 35" 34 y76" 48 34" 35 Slabs y11" 9 y5" 4 y1" 11 21" 10 y7" 6 Post-Glacial Rebound 1" 0.5 1" " 0.5 y0.2" 0.3 y3" 0.5 TECT-1 3"5 2"3 4"6 y1"4 y8"4 TECT-1rŽ ABC, QPS. 2.5" 2 2.5" 2 2.5" 2 y4" 3 y4" 3 TECT-1rŽ ABCQ, PS. 1.6" " " " 0.8 y8" 4 represented by TECT-1 is, among these choices, best represented by the two-region regionalization: Ž ABCQ, PS.. Although the projection of TECT-1 onto Ž ABCQ, PS. results in a variance reduction of only about 3%, this value exceeds those obtained from almost 95% of the projections associated with random rotations of the data sphere. This result is consistent with the roughly 2 s result associated with the platform and shield signal represented in TECT-1 Ž Table 6., but contrasts markedly with the results for the five-region grouping Ž A, B, C, Q, PS., where the actual orientation of the data sphere explains more of the variance than only 54% of the random orientations. This apparent discrepancy arises because random orientations of the other tectonic regions can lock on to regional features in the geoid 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. At these wavelengths Ž ls2 36., if there were no contribution from density contrasts at depths greater than 120 km, the geoid anomaly associated with isostatically compensated platforms and shields would be about q10 m, referenced to old ocean basins, or 0 m, referenced to ocean crust of zero age or to young continental crust Ž e.g., Hager, Our estimate of the geoid anomaly associated with old ocean basins, from the TECT-1 projections, is 4"6 m, for oceans 0 25 Ma is 3"5 m, and for young continents is y1" 4 m. Depending on whether we take old oceans, young oceans, or young continents as the reference value, our estimate of the signal due to the tectosphere alone, correcting for the effects of the crust, would be y22 m, y11 m, or y7 m. Because the old oceanic regions may still have some residual effect of subduction included in their estimate, and because the area-weighted average of young oceans and young continents is close to Table 7 Variance reductions and the corresponding confidence levels associated with the projection onto different groups of tectonic regions of five model geoids each representing a separate contribution to the geoid. Confidence level represents the percent of random trials that yield a smaller reduction in variance than that of the actual orientation of each geoid contributor Geoid Contributor Projection Variance reduction Ž %. Confidence Ž %. Upper 120 km A, B, C, Q, PS Lower Mantle A, B, C, Q, PS Slabs A, B, C, Q, PS Post-Glacial Rebound A, B, C, Q, PS TECT-1 A, B, C, Q, PS 4 54 TECT-1 ABC, QPS 2 78 TECT-1 ABCQ, PS 3 94

12 146 ( ) S.S. Shapiro et al.rlithos

13 ( ) S.S. Shapiro et al.rlithos zero, we retain the estimate of y8 m as the signal due to the continental tectosphere. 5. Estimate of lnrr lnn s The isostatic geoid height anomaly, d N, associated with static density anomalies can be calculated for each lateral location from Že.g., Haxby and Turcotte, 1978.: y2p G dns HD r Ž z. zd z Ž 4. g where G is the universal gravitational constant, g is the acceleration due to gravity, and D rž z. is the anomalous density at depth z. The integration extends from the surface to the assumed depth of compensation. Assuming that Eln rrelnns is constant within a specified depth interval, we may write the scaling there between fractional perturbations in density and shear-wave velocity as: D r E ln r Dt fy Ž 5. r E lnn t ž / s ž / where r is obtained, for example, from the radial earth model PREM ŽDziewonski and Anderson, 1981., and the fractional perturbations in shear wave velocity Dnsrns are equal to the negative of the fractional travel-time perturbations Dtrt, for small perturbations. We base the subsequent calculation on a depth of compensation of 440 km. Below this depth, we assume that there is no platform and shield contribution to the geoid, as there is no significant distinction at such depths between the shear-wave signal beneath platforms and shields and the global average Ž Shapiro, Using S12_WM13 Ž Su et al., 1994., we calculated regional averages of one-way shear-wave travel-time perturbations for 100-km-thick layers between 140 and 440 km depth. We then approximate the integral of the depth-dependent density anomaly Table 8 S12_WM13 Ž Su et al., Ž ls1 12.: Platform and shield averages and uncertainties corresponding to one-way S-wave travel-time anomalies Ž Shapiro, Ž. Ž. Ž. Depth interval km Dt rt % PS y2.3" y1.6" y1.0"0.2 as the sum of the anomalies for these layers. Using the travel-time perturbations Ž Shapiro, ŽTable 8. and dn PS 'g PS fy7.7"3.9 m, we find that for platforms and shields,the average value of Eln rrelnn is about 0.041" Ž s This estimate of standard error is based only on that of dn PS. The uncertainties associated with the regionally averaged travel-time perturbations have a much smaller effect on the value of Eln rrelnns than the uncertainty associated with the geoid and are therefore ignored.. 6. Discussion None of the projections based on Ž. 1 the non-hydrostatic geoid, Ž. 2 free-air gravity anomalies, or Ž. 3 our model geoid, TECT-1, yields a platform and shield signal that is significant at a level exceeding about 2.0s. Our conclusion is in accord with that reached by Doin et al. Ž using a geologic regionalization based on the tectonic map of Sclater et al. Ž They estimated that shields have a geoid difference from midoceanic ridges of between y10 m and 0 m; their corresponding estimate for platforms, which they keep as a separate region, is y4 m to 1 m, while they found essentially no difference in geoid for tectonically active continental areas and ridges. They were unable to estimate formal errors because of the previously discussed red nature of the spectra, but these values represent their subjective estimates of confidence intervals. Although 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 Hammer equal-area projection with coastlines drawn in white. Negative contour lines are dashed and the zero contour line is thick. The contour interval is 10 m.

14 148 ( ) S.S. Shapiro et al.rlithos their study and ours, their estimates fall within our uncertainties, and their conclusion that the tectosphere is compositionally distinct is consistent with ours. These observations differ substantially from the highly significant Ž 99% confidence. correlation, reported by Forte et al. Ž 1995., between an ocean continent function and the non-hydrostatic long-wavelength Ž l s 2 8. geoid. However, a correlation coefficient Ž r. between different fields defined on a sphere is only meaningful Žsubject to tests of significance. for fields with Ž significantly. non-white spectra if correlation coefficients are determined separately for each spherical harmonic degree of interest Ž Eckhardt, Given the appropriate number of degrees of freedom associated with the correlation, one can nonetheless estimate the confidence level corresponding to the assumption that the true correlation is zero. Therefore, we estimate the effective number of degrees of freedom in the analysis of Forte et al. Ž and, using this value, estimate the probability that the correlation which they obtained is significantly different from zero. Under the conditions outlined above, we can estimate the effective number of degrees of freedom using Student s t distribution. For uncorrelated fields, the quantity tsrwnrž1yr 2.x 1r2 can be described by Student s t distribution with n degrees of freedom Ž e.g., Cramer, 1946; see also O Connell, We create 10,000 degree-eight fields, each with the same spectral properties as the non-hydrostatic geoid, by randomly selecting coefficients from a uniform distribution and then scaling them degree-by-degree so that the power spectrum of each synthetic field matches that of the geoid. From these synthetic fields and an ocean-continent function derived from GTR1, we generate a collection of 10,000 correlation coefficients Ž Fig. 6a.. We then estimate n by minimizing the x 2 in the fit of Student s distribution to this set of correlation coefficients Ž Fig. 6b.. Fig. 6c,d,e demonstrate the sensitivity of the fits to the value of n. As shown, values of n which differ from the estimated value Ž ns30. by even 5 degrees of freedom, noticeably degrade the fit. The correlation coefficient corresponding to the geoid and Ž ABC, QPS. Ž l s 2 8. is y0.18. However, using the geoid and an ocean-continent function Ž ls2 8. derived from the 58=58 tectonic regionalization of Mauk Ž 1977., we obtained the same value Ž y0.28. as Forte et al. Ž With the regionalization of Mauk Ž 1977., simulations like those described above yield 31 as the estimate of the effective number of degrees of freedom. The significance levels of the correlations associated with the GTR1 and Mauk Ž ocean continent functions are, respectively, about 85% and 95%. The dominant degree-two term in the geoid governs this correlation and highlights a difficulty associated with attaching significance to the correlations between such fields. For example, if one considers only degrees ls3 8, the significance levels of the correlations associated with the GTR1 and Mauk Ž ocean continent functions reduce to about 55% and 60%, respectively, and hence indicate insignificant correlations. Our conclusion also differs substantially from that of Souriau and Souriau Ž who, using a Monte Carlo scheme based on random rotations of the data sphere with respect to the tectonic sphere, found that the non-hydrostatic geoid Ž l s correlates significantly Ž at the 95% confidence level. with the surface tectonics defined by Okal Ž The close geoid-tectonic association obtained by Souriau and Souriau Ž is partially related to the fact that the regionalization of Okal Ž includes subduction zones; the association between the geoid and this regionalization is a result of the strong geoid-slab correlation Ž e.g., Hager, Unlike our study, Souriau and Souriau Ž perform their projections in the spatial rather than in the spherical harmonic domain. After reproducing their results, we repeated Fig. 6. Ž. a Histogram of correlations Ž r. between an ocean continent function derived from GTR1 and 10,000 synthetic degree-eight fields each with the same spectral properties as the non-hydrostatic geoid. The shaded and unshaded arrows indicate the variance reductions Ž. 2 associated with the actual geoid and the maximum variance reduction, respectively. b x, calculated from the fit of Student s t distribution with the set of t s calculated from tsrwžnrž1yr 2.x 1r2, plotted as a function of the number of degrees of freedom Ž n.. The minimum 2 value of x corresponds with ns30. Histogram of values of t with Student s t distribution with n degrees of freedom superposed: Ž. c ns25, Ž. d ns30, and Ž. e ns35.

15 ( ) S.S. Shapiro et al.rlithos

16 150 ( ) S.S. Shapiro et al.rlithos their suite of projections in the spherical-harmonic domain. We found that the correlation between the long-wavelength geoid Ž l s and the regionalization of Okal Ž is significant at about the 98% confidence level, slightly higher than the result of Souriau and Souriau Ž of about 95% from a spatial-domain analysis. However, when we substitute a slab-residual model geoid ŽHager and Clayton, for the geoid, we find that the confidence level reduces to about 50%, indicating that the signal observed by Souriau and Souriau Ž is largely due to the correlation between slabs and the regionalization. The isopycnic hypothesis Ž Jordan, predicts a value of zero for Eln rrelnn s. This value is within 2.0s of our estimate and indicates that at this level of significance, the isopycnic hypothesis is consistent with the average geoid anomaly associated with platforms and shields. We can also estimate the value of Eln rrelnns by considering only thermal effects on density: Eln r Ž 1rr.Ž d rrdt. f Ž 6. Elnn Ž 1rn.Ž dn rdt. s s s Using a coefficient of volume expansion of 3=10 y5 y1 K, we make two estimates: Ž. 1 Eln rrelnns f 0.23, using dnsrdtfy0.6 m s y1 K y1 from McNutt and Judge Ž and an average upper-mantle shear y1 velocity of n f4.5 km s, and Ž. s 2 Eln rrelnnsf Ž. y4 y1 0.27, using ElnnsrET fy1.1 = 10 K from Nataf and Ricard Ž The average of these estimates is inconsistent at about the 10s level with the value of Eln rrelnns that we estimate for the conti- nental tectosphere. Hence, our analysis indicates that a simple conversion of shear-wave velocity to density via temperature dependence is inappropriate for the continental tectosphere and that one must consider compositional effects. Our conclusion could not differ more completely from that of Pari and Peltier Ž Ž henceforth PP., who claim that they can rule out the hypothesis that neutrally buoyant, compositionally distinct material exists beneath cratons. Based on a match to the peak amplitude of a severely truncated Ž l s 2 8. free-air gravity anomaly at one location ŽHudson Bay., they argue that 0.21-Ž Eln rrelnn. s , consistent with the thermal estimate above, and inconsistent at the 8 10s level with our estimate. However, there are several easily identifiable differences between their approach and ours. Most importantly, we use a geologic regionalization to define cratons. PP define cratons as any region, beneath either continents or margins, that has high inferred densities at 30 km depth in heterogeneity model S.F1.KrWM13 Ž Forte et al., This definition of craton is inappropriate for testing the composition of tectosphere for many reasons, including: Ž km depth beneath continents is generally within the lower crust, not within the proposed isopycnic region of the continental tectosphere; Ž. 2 model S.F1.KrWM13 is a heterogeneity model based on a weighted fit both to the gravity field and to the seismic data, assuming that density and velocity anomalies are proportional through assumed depthdependent values of Ž Eln rrelnn. s which vary be- tween 0.21 and 0.34 Ž Karato, In regions where the seismic coverage is not good, this assumption introduces a strong gravitational bias into model S.F1.KrWM13, making the use of this model in the inversion for Ž Eln rrelnn. s an example of circular logic; Ž. 3 The use of this hybrid model fails to identify the South African craton, a region with thick tectosphere Ž e.g., Su et al., 1994., as a craton. Ž 4. PP emphasize the value of the fit at Hudson Bay, while our study weights all regions of the globe equally. We also note that the estimate of the amount of this peak free-air gravity anomaly attributable to mantle structure is suspect due to contamination from postglacial rebound Ž Simons and Hager, In summary, given their approach, and their non-geologic definition of cratons, it is not surprising that PP find a different value for Ž Eln rrelnn. s than we do. Their value applies to the mantle beneath regions of inferred high-density lower-crust in a model determined from a joint inversion of gravity and seismic data. Our value of Ž Eln rrelnn. s applies to cratons defined by geological processes. In summary, to obtain realistic estimates of the significance of correlations between data fields defined on a sphere requires that one consider the spectra of the data fields so that the number of degrees of freedom can be determined appropriately. Our analysis demonstrates that the relationship between the long-wavelength geoid and the ocean continent function is tenuous. The large difference in

17 ( ) S.S. Shapiro et al.rlithos correlation that we obtain with different ocean continent functions further illustrates its insignificance. From error estimates that account for the redness in the geoid, gravity field, and tectonic regionalization spectra, we conclude that neither the geoid nor the free-air gravity has a platform and shield signal that differs significantly Ž 2 s. from zero. Additionally Ž see Shapiro, 1995., by considering regionally averaged shear-wave travel-time anomalies together with our model of the continental tectosphere s contribution to the geoid, we find that Eln rrelnns is about 0.04" Although this estimate is consistent at the 2.0s level with the isopycnic hypothesis of Jordan Ž 1988., the slightly positive estimate suggests that the decreased density associated with compositional buoyancy does not completely balance the increased density associated with low temperatures. We also note that convection calculations addressing the stability and dynamics of the continental tectosphere indicate that Eln rrelnns is likely to vary somewhat with depth ŽShapiro et al., 1999; see also Forte et al., Thus, our estimate is a weighted average of a quantity that may vary with position. Acknowledgements We thank P. Puster and G. Masters for computer code and T.A. Herring, P. Puster, W.L. Rodi, and M. Simons for helpful discussions. Richard J. O Connell provided a useful review. Figs. 1, 2 and 5 were created using the Generic Mapping Tools software Ž Wessel and Smith, We performed many of the calculations using the Guilford College Scientific Computation and Visualization Facility which was created with funds from a grant from the National Science Foundation Ž CDA This work was also supported by National Science Foundation grant EAR References Chase, C.G., Subduction, the geoid, and lower mantle convection. Nature 282, Cramer, H., Mathematical methods of statistics. Princeton Univ. Press, Princeton. Crough, S.T., Jurdy, D.M., Subducted lithosphere, hotspots and the geoid. Earth Planet. Sci. Lett. 48, Doin, M.-P., Fleitout, L., McKenzie, D., Geoid anomalies and the structure of continental and oceanic lithospheres. J. Geophys. Res. 107, Dziewonski, A.M., Anderson, D.L., Preliminary reference earth model Ž PREM.. Phys. Earth Planet. Inter. 25, Eckhardt, D.H., Correlations between global features of terrestrial fields. Math. Geol. 16, Forte, A.M., Dziewonski, A.M., Woodward, R.L., 1993a. Aspherical structure of the mantle, tectonic plate motions, nonhydrostatic geoid and topography of the core mantle boundary. In: Le Mouel, J.-L., Smylie, D.E., Herring, T. Ž Eds.., Dynamics of the Earth s Deep Interior and Earth Rotation. Geophys. Monogr. Ser. 72. AGU, Washington, DC, Forte, A.M., Peltier, W.R., Dziewonski, A.M., Woodward, R.L., 1993b. Dynamic surface topography: a new interpretation based upon mantle flow models derived from seismic tomography. Geophys. Res. Lett. 16, Forte, A.M., Woodward, R.L., Dziewonski, A.M., Joint inversions of seismic and geodynamic data for models of three-dimensional mantle heterogeneity. J. Geophys. Res. 99, Forte, A.M., Dziewonski, A.M., O Connell, R.J., Continent-ocean chemical heterogeneity in the mantle based on seismic tomography. Science 268, Gurnis, M., Comment on Dynamic surface topography: a new interpretation based upon mantle flow models derived from seismic tomography by A.M. Forte, W.R. Peltier, A.M. Dziewonski and R.L. Woodward. Geophys. Res. Lett. 20, Hager, B.H., Global isostatic geoid anomalies for plate and boundary layer models of the lithosphere. Earth Planet. Sci. Lett. 63, Hager, B.H., Subducted slabs and the geoid: constraint on mantle rheology and flow. J. Geophys. Res. 89, Hager, B.H., Clayton, R.W., Richards, M.A., Comer, R.P., Dziewonski, A.M., Lower mantle heterogeneity, dynamic topography and the geoid. Nature 113, Hager, B.H., Clayton, R.W., Constraints on the structure of mantle convection using seismic observations, flow models, and the geoid. In: Peltier, W.R. Ž Ed.., Mantle Convection, Plate Tectonics and Global Dynamics. Gordon & Breach, New York, Hager, B.H., Richards, M.A., Long-wavelength variations in Earth s geoid: physical models and dynamical implications. Philos. Trans. R. Soc. London, Ser. A 328, Haxby, W.F., Turcotte, D.L., On isostatic geoid anomalies. J. Geophys. Res. 83, Jordan, T.H., The continental tectosphere. Rev. Geophys. Space Phys. 13, Jordan, T.H., Composition and development of the continental tectosphere. Nature 274, Jordan, T.H., Global tectonic regionalization for seismological data analysis. Bull. Seismol. Soc. Am. 71, Jordan, T.H., Structure and formation of the continental tectosphere. J. Petrology, Special Lithosphere Issue, Kaula, W.M., Geophysical applications of satellite determinations of the earth s gravitational field. Space Sci. Rev. 7,