Download The continental tectosphere and Earth`s long

Lithos 48 Ž1999. 135–152
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 Ž1. a simple model of the continental
crust and oceanic lithosphere, Ž2. the lower mantle, Ž3. subducted slabs, and Ž4. 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 rrEln ns within the 140–440 km depth range is 0.04 " 0.02. A continental tectosphere with an isopycnic Žequal-density.
structure ŽEln rrEln ns s 0. enforced by compositional variations is consistent with this result at the 2.0 s level. Without
compositional buoyancy, the continental tectosphere would have an average Eln rrEln ns f 0.25, exceeding our estimate by
10 s . 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, 1975. and the lack of a strong correla)
Corresponding author.
Present address: Department of Physics, Guilford College,
Greensboro, NC 27410, USA
1
tion between continents and the long-wavelength
geoid Že.g., Kaula, 1967., Jordan Ž1975. proposed
that continents are Ž1. characterized by thick Ž; 400
km. thermal boundary layers ŽTBLs. which translate
coherently during lateral plate motions, Ž2. stabilized
against small-scale convective disruption by gradients in density due to compositional variations, and
Ž3. not observable in the long-wavelength gravity
0024-4937r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved.
PII: S 0 0 2 4 - 4 9 3 7 Ž 9 9 . 0 0 0 2 7 - 4
136
S.S. Shapiro et al.r Lithos 48 (1999) 135–152
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 ŽJordan, 1978.. Instead, Jordan Ž1978.
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 Žequal-density. hypothesis ŽJordan, 1988.
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, 1984. and with the distribution of hotspots ŽChase, 1979; Crough and Jurdy,
1980; Richards and Hager, 1988.. 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
Že.g., Richards and Engebretson, 1992.. 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, 1998.. 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, 1983.. 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 Ž1979.
concluded that there is no systematic difference in
the geoid signal between oceanic and continental
regions. But, Souriau and Souriau Ž1983. demonstrated that there is a significant correlation between
the geoid Žspherical harmonic degrees l s 3–12. and
the tectonic regionalization of Okal Ž1977.. From
degree-by-degree correlations Ž l s 2–20., Richards
and Hager Ž1988. observed a weak association between geoid lows and shields. On the other hand,
Forte et al. Ž1995. reported that the degree 2–8 geoid
correlates significantly Ž99% 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. Ž1993b. and Pari and Peltier Ž1996., 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 ŽForte 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, 1993.. On the other hand, Hager and Richards
Ž1989. and Forte et al. Ž1993b. 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. Ž1995. showed that they
could improve their fit to the geoid if they allowed
subcontinental regions to have a different proportion-
S.S. Shapiro et al.r Lithos 48 (1999) 135–152
Table 1
GTR1 ŽJordan, 1981.
Region
Oceans
A
B
C
Continents
Q
P
S
Definition
Young oceans Ž0–25 My.
Intermediate-age oceans
Ž25–100 My.
Old oceans Ž )100 My.
Phanerozoic orogenic zones
Phanerozoic platforms
Precambrian shields
and platforms
Fractional
area Ž%.
61
13
35
13
39
22
10
7
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 ŽJordan, 1981. ŽTable
1, Fig. 1. and the geoid, EGM96 ŽLemoine et al.,
137
1996., referred to the hydrostatic figure of Earth
ŽNakiboglu, 1982. ŽFig. 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 Ž1977. and
Okal Ž1977., as well as the ocean-continent function.
Because the geoid spectrum is red, with the rootmean-square Žrms. value of a coefficient of degree l
decreasing roughly as ly2 , 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 ly1 , 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.
138
S.S. Shapiro et al.r Lithos 48 (1999) 135–152
S.S. Shapiro et al.r Lithos 48 (1999) 135–152
subtracting other contributions from the geoid estimates. By combining the upper-mantle shear-wave
travel-time anomalies associated with platforms and
shields ŽShapiro, 1995. and the results from this
study, we estimate, with uncertainties, the average of
Eln rrEln ns within the depth range 140–440 km, and
compare our estimate with the isopycnic hypothesis
of Jordan Ž1988..
139
Table 2
Okal Ž1977.
Region
Definition
Fractional
area Ž%.
D
C
B
A
T
M
S
Ocean Ž0–30 My.
Ocean Ž30–80 My.
Ocean Ž80–135 My.
Ocean Ž )135 My.
Trenches and marginal seas
Phanerozoic mountains
Shields
12.0
30.1
12.3
2.5
10.9
11.6
20.4
2. Tectonic regionalization and inversion
GTR1 and regionalizations published by Okal
Ž1977. ŽTable 2. and Mauk Ž1977. ŽTable 3. contain
six, seven, and 20 regions, respectively. Both GTR1
and the regionalization of Mauk Ž1977. are defined
on a grid of 58 = 58 cells, whereas the model of Okal
Ž1977. is defined using 158 = 158 and 108 = 158
cells. The regionalization of Mauk Ž1977. 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 Žincluding marginal basins. are defined by
equal increments in the square root of crustal age:
0–25 My ŽA., 25–100 My ŽB., and ) 100 My ŽC.
and the continental regions are classified by their
generalized tectonic behavior during the Phanerozoic: Phanerozoic orogenic zones ŽQ., Phanerozoic
platforms ŽP., and Precambrian shields and platforms
ŽS.. Like GTR1, the oceanic regions of Mauk Ž1977.
are based largely on crustal age. However, the continental regions of Mauk Ž1977. are classified by age
rather than by their tectonic behavior. The more
complex parameterization associated with the regionalization of Mauk Ž1977. 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
Ž1977. and from GTR1 are consistent with each
other and only significant at slightly better than the
one-standard-deviation level. The regionalization of
Okal Ž1977. is limited in the accuracy of its designation of regions. For example, Okal Ž1977. labels the
entire continent of Antarctica a shield, whereas a
significant fraction Žf 1r3. is orogenic in nature.
Okal Ž1977. 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 nŽ n s 1, 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 ŽA., intermediate-age oceans ŽB., and old oceans ŽC. of GTR1,
into one region, and Q, P, and S, into another region,
we can create a two-component Žocean–continent.
tectonic regionalization ŽABC, QPS.. For much of
this analysis, we combine regions P and S into one
region ŽPS..
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 lnm representing the Ž l,m. harmonic of region n, and coefficient
d l m representing the Ž l,m. harmonic of the observed
Fig. 2. Ža. Geoid, l s 2–36 ŽEGM96; Lemoine et al., 1996., referred to the hydrostatic figure of Earth ŽNakiboglu, 1982.; Žb. Projection of
Ža. onto ŽA, B, C, Q, PS., and Žc. Residual: Ža–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.
S.S. Shapiro et al.r Lithos 48 (1999) 135–152
140
Table 3
Mauk Ž1977.
Region
Oceans
1
2
3
4
5
Definition
7
Anomaly 0–5 Ž0–10 My.
Anomaly 5–6 Ž10–20 My.
Anomaly 6–13 Ž20–38 My.
Anomaly 13–25 Ž38–63 My.
Late Cretaceous sea floor
Ž63–100 My.
Early Cretaceous sea floor
Ž100–140 My.
Sea floor older than 140 My
Continents
8
9
10
11
12
13
14
15
16
17
18
19
20
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
6
Fractional
area Ž%.
61.5
4.0
10.4
6.9
10.2
21.1
5.4
3.5
38.4
1.4
7.1
0.7
0.4
1.4
1.8
2.7
9.5
1.9
1.8
1.5
6.2
2.0
Žor model. geoid or gravity field, we use a leastsquares approach to solve:
R lnmgn s d l m
Ž 1.
Žsummation convention implied here and below. for
the regional averages, gn . We include the additional
constraint:
A n gn s 0
Ž 2.
where A n represents the surface area spanned by
region n. This constraint ensures that gn have a zero
Žweighted. average, as, by definition, do the geoidheight Žand free-air gravity. anomalies. The
weighted-least-squares solution can be written:
g s w RT WR x
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 Ž1998. 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,
1989.. 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 Žin our case unity. nor so
large Ž) 10 6 times the data weight. that the inversion becomes numerically unstable.
y1
RT Wd
R lnm
Ž 3.
where the values
and A n 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 l m and zero constitute the vector d.
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
S.S. Shapiro et al.r Lithos 48 (1999) 135–152
of 10,000 trials, we Ž1. randomly select an Euler
angle triple from a parent distribution in which all
orientations are equally probable and then, in accord
141
with the selected triple, rigidly rotate the sphere on
lm
which the data residuals Ž d res
s d l m y R nl mgn . are defined, with respect to the sphere on which the surface
Fig. 3. Non-hydrostatic geoid ŽEGM96, l s 2–36.: Histograms of parameter values Ža. gA , Ž b . g B , Ž c . g C , Žd. g Q , Že. 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˜l m Žsee text.. Gaussian distributions, determined by the standard deviation, mean, and area of each histogram, are
superposed. Žf. Histogram of variance reduction resulting from 10,000 random rotations of the data sphere, characterized by d l m , 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.
142
S.S. Shapiro et al.r Lithos 48 (1999) 135–152
tectonics are defined Ž‘‘tectonic sphere’’., Ž2. comlm Ž
bine the rotated data residuals d˜res
‘‘; ’’ denotes
rotated. with the correlated data to produce pseudo
lm
data, Ž d˜l m s d˜res
q R nl mgn ., and Ž3. project d˜l m onto
ŽA, B, C, Q, PS.. The resulting histograms of parameter values Že.g., gA , g B , g C , . . . . approximate
Fig. 4. Free-air gravity Ž l s 2–36.: Histograms of parameter values Ža. gA , Ž b . g B , Ž c . g C , Ž d . g Q , Ž 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˜l m Žsee
text.. Gaussian distributions, determined by the standard deviation, mean, and area of each histogram, are superposed. Žf. Histogram of
variance reduction resulting from 10,000 random rotations of the data sphere, characterized by d l m , 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.
S.S. Shapiro et al.r Lithos 48 (1999) 135–152
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 ŽFigs. 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 Žwhite 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
ŽShapiro, 1995.. 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 ŽShapiro, 1995. in the
limit of large numbers of trials with those determined from the elements of the variance vector
2
2
z ' xpost
diagwRT WRx -1 4 , where xpost
, is the Žpost2
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.,
2
2 .x
inversion, is thus defined by 100w1 y Ž xpost
rxpre
.
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.
143
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 ŽA, 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 ŽC. and ŽPS. 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% Ž2 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 ŽA, 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 ŽA, B, C, Q, PS. ŽFig. 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 ŽFig. 4a–e..
The significance of three of these averages is at or
below the 1.5s level and the significance of the
fourth, g Q , is at the 2.5s level ŽTable 4..
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 ŽFig. 3f., the free-air gravity projection explains more of the variance than about 98% of
projections corresponding with random orientations
Table 4
EGM96 Ž l s 2–36.: Regional averages and statistical standard errors from projections of the geoid and of perturbations to the free-air
gravity onto ŽA, B, C, Q, PS.
Geoid Žm.
Gravity Žmgal.
gA
gB
gC
gQ
g PS
0 " 12
4.0 " 3.4
1.8 " 5.6
y1.7 " 1.7
17 " 16
y4.8 " 3.9
y4.8 " 11
6.4 " 2.5
y10.5 " 8.3
y4.3 " 2.9
S.S. Shapiro et al.r Lithos 48 (1999) 135–152
144
of the data sphere ŽFig. 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 Ž1977. 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 Ž1977. into three groups based on crustal
age Žregions w1–7x, w8–14, 16–17x, w15, 18–20x.,
yields regional averages which are roughly the same
magnitude as their corresponding uncertainties ŽTable 5. and a variance reduction of about 6%. Another
continental grouping Žw1–7x, w8–10, 12–13x, w11, 14–
Table 5
EGM96 Ž l s 2–36.: Regional averages and statistical standard
errors from projections of the geoid onto several regionalizations
based on Mauk Ž1977.. Group 1: Žw1–7x, w8–14, 16–17x, w15,
18–20x.; Group 2: Žw1–7x, w8–10, 12–13x, w11, 14–17x, w18–20x.;
Group 3: 20 separate regions
Region
Oceans
1
2
3
4
5
6
7
Continents
8
9
10
11
12
13
14
15
16
17
18
19
20
Group 1
g Žm.
Group 2
g Žm.
9"6
9"6
9"6
9"6
9"6
9"6
9"6
9"6
9"6
9"6
9"6
9"6
9"6
9"6
y22"16
y22"16
y22"16
y22"16
y22"16
y22"16
y22"16
y7"9
y22"16
y22"16
y7"9
y7"9
y7"9
y9"15
y9"15
y9"15
y19"14
y9"15
y9"15
y19"14
y19"14
y19"14
y19"14
y12"13
y12"13
y12"13
Group 3
g Žm.
y13"14
10"14
9"8
5"7
4"10
15"14
40"25
127"46
y41"19
62"44
y10"57
27"41
6"37
y34"22
4"14
y61"33
y60"26
37"28
y27"15
y23"23
17x, w18–20x. based instead on a combination of age
and tectonic behavior, yields similar Žinsignificant.
results ŽTable 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 Ž1977., the large
relative uncertainties Žand even differences in sign.
associated with old continents ŽTable 5. suggest that
this association is indeed weak. In addition, there are
only three regions Ž8, 9, and 17. that have average
values that differ from zero by more than 2 s .
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 Žnon-hydrostatic. geoid
the effects of previously modeled components: Ž1. 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 2–20. ŽHager,
1983.; Ž2. the lower mantle Ž l s 2–4. ŽHager and
Clayton, 1989.; Ž3. slabs Ž l s 2–9. ŽHager and Clayton, 1989.; and Ž4. remnant glacial isostatic disequilibrium Ž l s 2–36. ŽSimons and Hager, 1997.. Separately projecting each of these four contributions to
the model geoid onto ŽA, B, C, Q, PS. yields the
results given in Tables 6 and 7. Our resulting model
Žresidual. geoid, TECT-1 ŽFig. 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 PS f -8 " 4 m ŽTable
6..
The projections of TECT-1 separately onto ŽA, B,
C, Q, PS., ŽABC, QPS., and ŽABCQ, 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
Žconfidence level., it is clear that the geoid signal
S.S. Shapiro et al.r Lithos 48 (1999) 135–152
145
Table 6
Regional averages and statistical standard errors from projections onto ŽA, 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 ŽABC, QPS. and ŽABCQ, PS.
Geoid contributors
gA Žm.
g B Žm.
g C Žm.
g Q Žm.
g PS Žm.
Upper 120 km
Lower Mantle
Slabs
Post-Glacial Rebound
TECT-1
4.3 " 1.0
y5 " 32
y11 " 9
1 " 0.5
3"5
y3.1 " 0.5
20 " 19
y5 " 4
1 " 0.3
2"3
y6.1 " 1.0
35 " 34
y1 " 11
0.7 " 0.5
4"6
3.1 " 0.9
y76 " 48
21 " 10
y0.2 " 0.3
y1 " 4
3.5 " 0.8
34 " 35
y7 " 6
y3 " 0.5
y8 " 4
TECT-1rŽABC, QPS.
TECT-1rŽABCQ, PS.
2.5 " 2
1.6 " 0.8
2.5 " 2
1.6 " 0.8
2.5 " 2
1.6 " 0.8
y4 " 3
1.6 " 0.8
represented by TECT-1 is, among these choices, best
represented by the two-region regionalization:
ŽABCQ, PS.. Although the projection of TECT-1
onto ŽABCQ, 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 ŽTable 6., but contrasts markedly with the
results for the five-region grouping ŽA, 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.
y4 " 3
y8 " 4
At these wavelengths Ž l s 2–36., 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 Že.g., Hager, 1983..
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
Lower Mantle
Slabs
Post-Glacial Rebound
TECT-1
A, B, C, Q, PS
A, B, C, Q, PS
A, B, C, Q, PS
A, B, C, Q, PS
A, B, C, Q, PS
80
37
13
19
4
100
88
77
100
54
TECT-1
TECT-1
ABC, QPS
ABCQ, PS
2
3
78
94
146
S.S. Shapiro et al.r Lithos 48 (1999) 135–152
S.S. Shapiro et al.r Lithos 48 (1999) 135–152
zero, we retain the estimate of y8 m as the signal
due to the continental tectosphere.
5. Estimate of ≥ lnr r ≥ lnns
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.:
dNs
y2p G
g
HD r Ž z . zd z
Ž 4.
where G is the universal gravitational constant, g is
the acceleration due to gravity, and D r Ž z . is the
anomalous density at depth z. The integration extends from the surface to the assumed depth of
compensation. Assuming that Eln rrEln ns is constant
within a specified depth interval, we may write the
scaling there between fractional perturbations in density and shear-wave velocity as:
Dr
r
fy
E ln r
Dt
E ln ns
t
ž /ž /
147
Table 8
S12_WM13 ŽSu et al., 1994. Ž l s1–12.: Platform and shield
averages and uncertainties corresponding to one-way S-wave
travel-time anomalies ŽShapiro, 1995.
Depth interval Žkm.
Ž Dt rt . PS Ž%.
140–240
240–340
340–440
y2.3"0.2
y1.6"0.2
y1.0"0.2
as the sum of the anomalies for these layers. Using
the travel-time perturbations ŽShapiro, 1995. ŽTable
8. and d NPS ' g PS f y7.7 " 3.9 m, we find that for
platforms and shields,the average value of
Eln rrEln ns is about 0.041 " 0.021. ŽThis estimate of
standard error is based only on that of d NPS . The
uncertainties associated with the regionally averaged
travel-time perturbations have a much smaller effect
on the value of Eln rrEln ns than the uncertainty
associated with the geoid and are therefore ignored..
6. Discussion
Ž 5.
where r is obtained, for example, from the radial
earth model PREM ŽDziewonski and Anderson,
1981., and the fractional perturbations in shear wave
velocity D nsrns 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 ŽShapiro, 1995..
Using S12_WM13 ŽSu 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
None of the projections based on Ž1. the non-hydrostatic geoid, Ž2. free-air gravity anomalies, or Ž3.
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. Ž1996. using a geologic
regionalization based on the tectonic map of Sclater
et al. Ž1980.. 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
Fig. 5. Ža. TECT-1, l s 2–36; Žb. Projection of Ža. onto ŽA, B, C, Q, PS., and Žc. Residual: Ža–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.
148
S.S. Shapiro et al.r Lithos 48 (1999) 135–152
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 Ž99% confidence. correlation, reported by Forte et al. Ž1995., 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 Žsignificantly. non-white spectra if correlation coefficients are determined separately for each spherical harmonic degree of interest
ŽEckhardt, 1984.. 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. Ž1995. 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 t s r w nrŽ1 y r 2 .x1r2 can be described
by Student’s t distribution with n degrees of freedom Že.g., Cramer, 1946; see also O’Connell, 1971..
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 ŽFig. 6a.. We then estimate n by minimizing
the x 2 in the fit of Student’s distribution to this set
of correlation coefficients ŽFig. 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 n s 30. by even 5 degrees of freedom, noticeably degrade the fit.
The correlation coefficient corresponding to the
geoid and ŽABC, QPS. Ž l s 2–8. is y0.18. However, using the geoid and an ocean-continent function Ž l s 2–8. derived from the 58 = 58 tectonic regionalization of Mauk Ž1977., we obtained the same
value Žy0.28. as Forte et al. Ž1995.. With the
regionalization of Mauk Ž1977., 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 Ž1977. 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 l s 3–8,
the significance levels of the correlations associated
with the GTR1 and Mauk Ž1977. 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 Ž1983. 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 3–12. correlates significantly Žat the 95% confidence level. with the
surface tectonics defined by Okal Ž1977.. The close
geoid-tectonic association obtained by Souriau and
Souriau Ž1983. is partially related to the fact that the
regionalization of Okal Ž1977. includes subduction
zones; the association between the geoid and this
regionalization is a result of the strong geoid-slab
correlation Že.g., Hager, 1984.. Unlike our study,
Souriau and Souriau Ž1983. perform their projections
in the spatial rather than in the spherical harmonic
domain. After reproducing their results, we repeated
Fig. 6. Ža. 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
associated with the actual geoid and the maximum variance reduction, respectively. Žb. x 2 , calculated from the fit of Student’s t distribution
with the set of t’s calculated from t s r wŽ nrŽ1 y r 2 .x1r 2 , plotted as a function of the number of degrees of freedom Ž n .. The minimum
value of x 2 corresponds with n s 30. Histogram of values of t with Student’s t distribution with n degrees of freedom superposed: Žc.
n s 25, Žd. n s 30, and Že. n s 35.
S.S. Shapiro et al.r Lithos 48 (1999) 135–152
149
S.S. Shapiro et al.r Lithos 48 (1999) 135–152
150
their suite of projections in the spherical-harmonic
domain. We found that the correlation between the
long-wavelength geoid Ž l s 3-12. and the regionalization of Okal Ž1977. is significant at about the 98%
confidence level, slightly higher than the result of
Souriau and Souriau Ž1983. of about 95% from a
spatial-domain analysis. However, when we substitute a slab-residual model geoid ŽHager and Clayton,
1989. for the geoid, we find that the confidence level
reduces to about 50%, indicating that the signal
observed by Souriau and Souriau Ž1983. is largely
due to the correlation between slabs and the regionalization.
The isopycnic hypothesis ŽJordan, 1988. predicts
a value of zero for Eln rrEln ns . This value is within
2.0 s 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 rrEln ns by considering only thermal effects on
density:
Eln r
f
Eln ns
Ž 1rr . Ž d rrdT .
Ž 1rns . Ž dnsrdT .
Ž 6.
Using a coefficient of volume expansion of 3 = 10y5
Ky1 , we make two estimates: Ž1. Eln rrEln ns f 0.23,
using dnsrdT f y0.6 m sy1 Ky1 from McNutt and
Judge Ž1990. and an average upper-mantle shear
velocity of ns f 4.5 km sy1 , and Ž2. Eln rrEln ns f
0.27, using ŽEln nsrET . f y1.1 = 10y4 Ky1 from
Nataf and Ricard Ž1996.. The average of these estimates is inconsistent at about the 10 s level with the
value of Eln rrEln ns that we estimate for the continental 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 Ž1996. Žhenceforth 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 - ŽEln rrEln ns . - 0.26,
consistent with the thermal estimate above, and in-
consistent at the 8–10 s 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 ŽForte et al., 1994.. This definition
of ‘‘craton’’ is inappropriate for testing the composition of tectosphere for many reasons, including: Ž1.
30 km depth beneath continents is generally within
the lower crust, not within the proposed isopycnic
region of the continental tectosphere; Ž2. 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 ŽEln rrEln ns . which vary between 0.21 and 0.34 ŽKarato, 1993.. 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 ŽEln rrEln ns . an example of circular
logic; Ž3. The use of this hybrid model fails to
identify the South African craton, a region with thick
tectosphere Že.g., Su et al., 1994., as a craton. Ž4. 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 ŽSimons and Hager, 1997.. In summary, given their approach, and their non-geologic
definition of cratons, it is not surprising that PP find
a different value for ŽEln rrEln ns . 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 ŽEln rrEln ns . 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
S.S. Shapiro et al.r Lithos 48 (1999) 135–152
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 Ž2 s . from zero. Additionally
Žsee 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 rrEln ns is about
0.04 " 0.02. Although this estimate is consistent at
the 2.0 s level with the isopycnic hypothesis of
Jordan Ž1988., 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 rrEln ns is likely to vary
somewhat with depth ŽShapiro 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 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
ŽWessel and Smith, 1991.. 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 ŽCDA-9601603.. This work was
also supported by National Science Foundation grant
EAR-9506427.
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