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JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 85, NO. B I 1, PAGES 6377-6396, NOVEMBER 10, 1980

Implications of Regional Gravity for State Stress in the Earth's Crust and Upper Mantle

MARCIA MCNUTT

U.S. Geological Survey, Menlo Park, California 94025

Topography is maintained by stress differences within the earth. Depending on the distribution of the stress we classify the support as either local or regional compensation. In general, the stresses implied in a regional compensation scheme are an order of magnitude larger than those corresponding to local isostasy. Gravity anomalies, a measure of the earth's departure from hydrostatic equilibrium, can be used to distinguish between the two compensation mechanisms and thus to estimate the magnitude of deviatoric stress in the crust and upper mantle. Topography created at an oceanic ridge crest or in a major continental orogenic zone appears to be locally compensated. Such features were formed on weak crust incapable of maintaining stress differences much greater than the stress from the applied load. Oceanic volcanoes formed on an already cooled, thickened lithosphere are regionally supported with elastic stresses. Simi- larly, the broad topographic rise seaward of subduction zones is elastically supported as the lithosphere is bent near the plate margin. Although the implied stress is to some degree dependent on the theological model assumed, the gravity anomalies and surface deformation produced by these features demonstrate that the upper 30-40 km of the oceanic lithosphere is capable of regionally supporting stress differences in the 100-MPa range. Given certain conditions of load emplacement, continental crust can also support loads regionally over 100-m.y. time scales, but the effects of erosion only allow an estimate of a lower bound on stress. Data from space probes indicate that the upper layers of other terrestrial planets also support topographic-induced stress differences in excess of 100 MPa.

INTRODUCTION

Seismologists tell us that above the core the mantle reacts as an elastic solid to the passage of seismic disturbances. The effective seismic rigidity, or resistance to deformation by shearing stresses, is comparable to the strength of steel, but this estimate only applies to short-period stresses of relatively small magnitude. The strength may be much smaller for large stresses of long duration, and in general, the problem dis- cussed here is one of determining the earth's 'permanent' resistance to shearing stresses as a function of depth, amplitude, and wavelength of the disturbance. This question is fundamental in that it bears on many other geophysical problems, such as the development of mountain chains, long-term vertical motions of the earth's surface, the temperature structure in the crust and uppermost mantle, the scale and rate of convection, and other aspects of earth dynamics.

From an historical viewpoint the first estimates of stress in the earth were based on gravity anomalies. The fact that stress differences exist cannot be denied. Topographic features alone represent a departure from hydrostatic equilibrium, and the earth's reaction to the surface load according to certain rheological laws provides a means of distributing the stresses over depth within the earth. Gravity anomalies are most often used to estimate the earth's response to surface stress, and thus the stress issue naturally becomes involved with the question of isostatic compensation. The discussion of stress within the earth must not be limited to that which is topographica,Hy induced; gravity anomalies also tell us that there exis( mass anomalies, and therefore deviatoric stresses, that are unrelated to existing surface elevations.

The purpose of this paper is to review estimates of earth strength based on gravity observations, although consideration will also be given to studies of isostatic compensation based on surface deformation. The prevailing theme that emerges is that there exists a region in the crust and up-

This paper is not subject to U.S. copyright. Published in 1980 by the American Geophysical Union.

permost mantle that supports significant deviatoric stress, a 'lithosphere.' Note that the thickness of the lithosphere can be defined on the basis of seismic, thermal, compositional, or mechanical properties. We will use the term in the restricted sense of the mechanical lithosphere, that portion of the crust and upper mantle with long-term strength.

Within the context of this discussion we define 'isostasy' as the condition in which all stresses are hydrostatic below some compensation depth. This definition encompasses both local and regional compensation mechanisms. A feature is said to be locally compensated if the total mass in any vertical col- umn above the compensation depth is constant. In this scheme, elevated regions are pointwise compensated. A regional mechanism distributes compensation laterally around the feature as well as vertically beneath it, implying that shear stresses can be transmitted horizontally.

We will begin with a brief discussion of Jeffreys' [1924, 1943, 1976] work, because his concepts have in some way influenced the thoughts of almost all subsequent investigators. The second section concentrates on strength estimates derived from measurements of the longest-wavelength components of the earth's gravity field. The following two sections deal with oceanic and continental studies of a more regional dimension. Finally, the results for the earth are compared with stress estimates for the moon and Mars based on recent elevation and

gravity measurements by space probes.

JEFFREYS' WORK

Given only a surface distribution of topography, Jeffreys [1976] describes three approaches to determining stress:

1. Assume an earth rheology and mechanism for isostatic compensation. Apply the load, allow the earth to respond, and calculate the stress.

2. Assume the dynamic processes that form the topography, and work out the stress consequences.

3. Calculate all possible stress distributions consistent with the surface load. The distribution that attains the least maxi-

Paper number 80B0277. 6377

6378 MCNUTT: IMPLICATIONS OF GRAVITY FOR STRESS

T

i

,

ø-mex :.Kn $n

h h0

- •---a0uguer An•mol• . _•(elestic pluto) '•'"•"••• TI TITI• .... f • • Pm Asthen0sphere

Buoyancy Forces (fluid

O-max--up t0 I0 hg/o/e

- 500 ioo .12 % 0 0 , ,

..... + ..... 50

Crmo x = hgp/e I

ot z: • Tr x w0velength of 10ud

Airy Compensetion• •

O-ma x; hgp Fig. 1. Models used by Jefftoys [1976] to calculate stress in the earth from surface loading. (a) Harmonic loading on an

elastic earth. Maximum stress (amax) depends on the root-mean-square of the topographic stress (s,) and a multiplicative factor (k,), which depends only on harmonic degree. (6) Harmonic loading on a flat earth. Orientation of the stress ellipsoid is also shown. (c) Loading on an elastic plate overlying a fluid. From Bank• et M. [1977], reprinted with permission of Blackwell Scientific Publications. (g) Airy compensation.

mum stress difference provides a lower bound for the strength of the earth.

The first two methods are examples of the 'forward' problem in geophysics; the stress answers are no more valid than the assumptions. The first method is often used to test the plausibility of various rheologies, using predicted stress levels, gravity anomalies, and surface deformation as a measure of the model's acceptability. The second approach is rarely used owing to a lack of information on the fundamentals of orogeny. The third type of analysis represents the true 'inverse' problem in that the results depend only on the observations, not on the assumptions. It should be remembered that the bound itself is the only quantity of importance in the inverse approach. The stress distribution that attains the least maximum stress may not resemble that of the earth, since it was not required that the solution result from any known or even plausible earth behavior.

TABLE 1. Stress From Loading on an Elastic Earth

n k. s•, MPa Ao,,, MPa r/a Depth, km

2 3.03 21.3 64.6 0 6400 3 1.70 23.4 39.8 0.591 2600

10 1.13 9.6 10.8 0.895 670 30 1.025 3.4 3.4 0.966 220 oo 1.040

10 6 Pa = 10 bars.

Jeffreys begins with the first approach, assuming an elastic rheology for the earth. This is basically a Bouguer theory cor- rected for earth elasticity and makes no assumption of isostatic compensation by requiring that the stress be hydrostatic below some depth. The value of this approach is twofold. An analytic solution can be easily obtained, thus avoiding tedious numerical calculation of stress distributions. Secondly, as Jef- freys argues, the elastic theory uncorrected for isostasy gives a better estimate of the lower bound on the stress differences be-

cause the entire earth contributes to the support of the load. If we assume that the interior of the earth is hydrostatic and thus supports no stress differences below a certain depth, the maximum stress difference in the overlying elastic layer must increase to compensate for the loss of support from below. Jef- freys also finds that the least maximum stress difference given by the elastic theory is not much more than the value determined by the inverse approach. The reason for this perhaps surprising result is that any nonelastic solution for stress within the earth must increase the total strain energy per unit volume (Castigliano's principle) even if it does reduce the maximum deviatoric stress. Thus while the inverse approach finds stress solutions that decrease the maximum of a func-

tion, it does so at the expense of increasing the volume integral of that same function. The result is that stress values nearly comparable to the maximum are spread over a greater volume.

MCNUTT: IMPLICATIONS OF GRAVITY FOR STRESS 6379

The first case considered by Jeffreys is that of a surface harmonic load resting on an incompressible elastic sphere (Figure la). Let sn equal the root-mean-square variation of the topographic stress of degree n as given in terms of fully normalized spherical harmonics. The maximum stress difference in the sphere from the applied normal stress is given by kns, in which k,is a multiplicative factor dependent only on n and is located at a depth of approximately 1/n times the sphere's radius. Using the values for k,and depth from Jeffreys [1943] and the topography coefficients for the earth from Balmino et al. [1973], we can construct Table 1 of stress differences Ao, defined as the difference between the maximum and mini-

mum principle stresses, and distance from the center of the earth in fraction of earth radius that the stress maximum oc-

curs. The numbers in this table reveal a theme that reappears in many later studies. For degrees higher than n -- 10 the maximum stress difference is less than 10 MPa and occurs at

decreasing depths. Although estimates of the strength of the earth's uppermost regions vary [Lainbeck, 1972], the 10 MPa necessary to support the topographic stress assuming an elastic rheology is well within even the most conservative estimates. The low-order harmonics do present a problem, however. While it is highly probable that rocks withstand stress differences at least as targe as 70 MPa in crustal environments, it is unlikely that the requisite strength is present in the mantle between the lithosphere and the core. If we require that the support for the low-order harmonics lie in the lithosphere, the maximum stress difference can increase to several hundred

megapascals. For features less than a few hundred kilometers in horizon-

tal scale, Jeffreys adopts a flat earth model, shown in Figure lb. Given a normal stress distribution with harmonic form

p• = t•gh cos (•x)

where p is the load density, the maximum stress difference is of the order of 2pgh/e and occurs at a depth of 1/2•r times the wavelength of the load. The stress solutions for loads modeled as raised strips with rectangular or triangular section are similar; in general, the greatest stress difference in the elastic theory is between one half and two thirds of the range of the load stress, and the greatest strength is needed at a depth about one quarter of the width of the load. Applying these approximate equations to the Himalayas, for which Jeffreys assumes a height difference of 5 km between peaks and troughs and a chain width of 100 km, he estimates that stress differences reach 450 MPa at depths of 20-30 km if the mountains are elastically supported.

Jeffreys' flat earth treatment can be modified to include the effects of isostasy by assuming that the elastic support resides only in a thin elastic plate overlying a fluid that can support no shear stresses (Figure lc). For loads with small amplitude and wavelength less than Te, the thickness of the plate, the above solution approximately holds because the region of appreciable stresses resides above the fluid. For loads whose wavelength is long in relation to the thickness of the elastic layer, the thin-plate equations from elastic plate theory can be applied. Jeffreys considers a case in which the plate is 50 km thick. The flexural rigidity

ETe 3 D = 12(1- v 2) (1)

where E is Young's modulus and v is Poisson's ratio, corresponding to this Te value is 9 x 10 •3 N m. The support from

the plate leads to departures from perfect local isostasy for short-wavelength loads, but longer-wavelength loads are nearly locally compensated. For a 1-km-amplitude harmonic load of density 2500 kg/m 3 and wavelength 450 km the maximum stress difference in the plate reaches 300 MPa. In general, for a floating crust the maximum stress may be as much as 10 times the amplitude of the surface stress. Jeffreys concludes that if mountains are supported by a floating elastic plate, laboratory measurements on the strength of rocks indicate that long-wavelength inequalities 5 km in height should cause fracture of the crust.

Taking again the case in which strength is uniform with depth, nonelastic solutions do not reduce the maximum stress significantly. The greatest improvement is found for harmonic loads of degree 2 and 3, for which the reduction in necessary strength is from 5 to 30%. For two-dimensional harmonic loading on a planar boundary, however, the reduction in the maximum stress difference from the elastic solution is only about 7%.

A familiar example of a nonelastic solution for a floating crust is the Airy isostatic mechanism, shown in Figure ld. Compensation for surface features is achieved by thickening the •:rust below elevated regions. For perfect isostatic equilibrium the vertical pressure from the load is balanced by the buoyancy pressure from fluid displaced by a root of depth w: wgAp ----- hgp, where Ap is the density difference between the fluid and the crust and h the land elevation of density p. The maximum stress difference in the crust for this compensation distribution is equal to the magnitude of the load. For small- wavelength features the elastic solution described above gives a smaller stress maximum, but for wavlengths longer than 2.6 Te the Airy mechanism is optimal. Therefore the local isostasy statement that mass per unit area is constant does not lead to the smallest maximum stress differences in all cases. If

the definition of isostasy is amended to require that the maximum stress difference be minimized and in addition that all

stresses be hydrostatic below the compensation depth, then the Airy mechanism would not be acceptable. However, free air gravity anomalies in the minimum stress state would be larger than what is actually observed. Although the Airy mechanism better agrees with observations because the compensation is total and extends to all wavelength features, the model is unrealistic. The requirement that small changes in the load produce vertical motion of crustal blocks that rigidly oppose any horizontal movement implies that crustal material is infinitely anisotropic to deformation.

Artyushkov [1973, 1974] has elaborated on this point. While local isostasy requires only that vertical forces equilibrate, the true stable position of the crust balances vertical forces, horizontal forces, and all moments. Deviatoric stresses cannot be homogeneously distributed over depth if a layer is locally compensated. Departures of a locally compensated crust from true equilibrium are inversely proportional to the characteristic horizontal scale of the topography or density in- homogeneity. Thus appreciable displacements of the crust from isostasy will occur only for very narrow features.

Given only the horizontal and vertical structure of lithospheric density inhomogeneities, Artyushkov [1973] devises a method of estimating deviatoric stress averaged over the lithospheric thickness. These stresses arise from lateral density and thickness inhomogeneities in an isostatically compensated crust. The buoyancy force at any density discontinuity is ented along the normal to the interface, which may not coin- cide with the vertical direction. For sloping interfaces, only


TABLE 2. Artyushkov's Stress Estimates

Feature Stress, MPa T•, km

Midocean Ridge Crest - 24 50

Margin (East Pacific Rise) - 55 50 Margin (Mid-Atlantic Ridge) -140 50

Continents

Margin 10 80 3-km uplift, supported by

crustal root 60 80

5-km uplift, supported by crustal root 110 80

2-km uplift, supported by low-density mantle 200 80

the vertical component of the buoyancy force is balanced by the weight of the topography, leaving an unbalanced horizontal component. For several examples of areas with strong crustal structure, estimates of the average deviatoric stress are given in Table 2.

Note that only the product of average stress times lithospheric thickness Te is determined by Artyushkov's analysis. The results in Table 2 may underestimate the deviatoric stress for the assumed models, since it is unlikely that the earth supports large stress differences below 40-50 km. The assumed density structure is also critical to Artyushkov's calculation. However, with deep seismic sounding and gravity data it is possible to resolve some of these details of crustal structure.

Further data are needed to determine the actual stress dis-

tribution within the lithosphere. In particular, the tectonic implications of the stresses in Table 2 depend primarily on the horizontal and vertical viscosity distribution in the lithosphere, which we are only beginning to understand. Never- theless, Artyushkov [1973] proposes a scheme of global tectonics in which horizontal motions are driven by lateral spreading of crustal density and thickness inhomogeneities. Continued differentiation of the core releases buoyant material that ascends through the mantle, forming the roots for new uplifts. Thus chemical differentiation provides the driving mechanism. At present there is little support for Artyushkov's proposal; since it is not possible to fix all ridge crests with respect to the lower mantle, it becomes difficult in Artyushkov's scheme to explain the continued existence of individual ridge systems over long geologic time spans. Many other mechanical, geochemical, and observational objections could be listed, but they do not directly bear on the results of Table 2. The fact remains that in regions of relief on internal density interfaces as well as on the earth's surface, deviatoric stresses exist,

and they are of the order of a hundred megapascals.

GLOBAL STUDIES

The global gravity field reveals departures from hydrostatic equilibrium that extend over thousands of kilometers. If these features are supported in an isostatic sense by the lithosphere, their continued existence implies significant finite strength. There is, however, a dynamic alternative to static support; the anomalies could be maintained by flow in the mantle. On the assumption that the lithosphere is the only region of the earth with appreciable strength, depth of the causative mass can be used as the criterion for distinguishing between these two pos- sibilities. Evidence from glacial rebound, isostatic compensation, and seismic studies confirms the existence of an extremely weak asthenosphere underlying the lithosphere.

While it is not yet possible to discount finite strength entirely in the lower mantle, the attenuation of upward-continued potential fields makes it unlikely that any but the longest-wavelength anomalies would have source depths greater than 900 km. Therefore if the gravity anomalies originate below the lithosphere, they more likely pertain to the convection problem. Any gravity anomalies that do reside within the lithosphere may provide an estimate of its strength, but the possibility still remains that mass anomalies located within the lithosphere are dynamically maintained by vertical forces at its base.

A method adopted by several investigators [Cruier and New- ton, 1965; Allen, 1972; Khan, 1977] for determining source depths for low degrees of the global gravity field is based on the assumption that the gravity anomalies are produced by randomly distributed density variations. The assumption of a white spectrum for the density variations is sufficiently re- strictive to permit a unique inversion for either the depth from which the density anomalies should extend uniformly down- ward or the depth of a single-density interface concentrating the anomalies. The results are dependent on the particular set of potential coefficients used [Higbie and Stacey, 1971], but a recent analysis by Khan [1977] shows that the latest satellite and combination solutions, WGS 72, Gem 7, Gem 8, and PGS 110, are consistent to degree 10 or 11.

Assuming a single-density interface, the source depths according to Khan [1977] are 600-800 km for n -- 2 to n -- 11 and 300-600 km for n -- 11 to n -- 30. Shallower depths are found on the assumption of a disordered mantle below the specified depth: 150-370 km for n -- 2 to n -- 11 and 150-450 km for n -- 11 to n -- 30. Only the anomalous part of C2 ø relative to the best fitting satellite reference ellipsoid was used in the analysis. The nonhydrostatic part of C2 ø is about an order of magnitude greater than the next largest coefficient and tends to bias the third- and fourth-degree components toward greater depth.

We could conclude from Khan's analysis that the major contribution to the global gravity field is from sublithospheric depth and therefore must be maintained by convection. How- ever, Goodacre [1978] has questioned the assumption that density variations are uncorrelated within the earth. The fact that the source depth depends on n may reflect the existence of several warped discontinuity surfaces, but it also may result from a red-shifted spectrum of density variations. To illustrate this point, Goodacre [1978] considers the earth's surface topography. The amplitude spectrum is not white; it varies as (2n + 1) -•/: [Balmino et al., 1973]. Using the potential coefficients for the gravity field from the topography, the depth estimate for the topography on the assumption that the density variations have a fiat spectrum is a few hundred kilometers, rather than the expected zero depth. There is no reason to be- lieve that internal density variations are any more random than those at the surface, and therefore the above depth estimates may be meaningless. Overall, this approach to interpreting the global gravity field is not highly promising, since it involves virtually untestable assumptions concerning the sta- tistical behavior of inhomogeneities deep within the mantle.

While the above procedure was designed to directly estimate source depth for gravity anomalies, McKenzie [1967] posed the problem in the reverse manner: If we suppose that gravity anomalies are supported by the lithosphere, then what stresses are implied? McKenzie then rejects a lithospheric source for anomalies that require stresses above an assumed


TABLE 3. Lambeck's Stress Estimates

n' Omax, MPa

4 875

5 487 6 313

7 219

8 166 9 129

10 103 11 84

12 70 13 , 59 14 49

15 39

16 31

ultimate strength. He restricts discussion to gravity anomalies that are produced by uncompensated warping of an elastic lithosphere, which could reasonably apply to subducting plates and broad lithospheric swells. Although it might be ar- gued that these large-scale plate deformations are a consequence of mantle flow, the only concern here is whether the lithosphere can maintain the configuration once it has been established.

From observed gravity and geoid anomaly amplitudes and wavelengths, McKenzie calculates a minimum stress of 83 MPa to support plate flexure near the Tonga and Puerto Rico trenches, assuming a 50-km-thick lithosphere. For a 100-km- thick plate the stress estimate is reduced to 22 MPa. The deci- sion as to whether the support for these features is derived from the lithosphere hinges on the choice for its ultimate strength. McKenzie cites laboratory experiments by Griggs et al. [1960] that produce shear failure in dunite at 400-MPa stress. He rejects this strength estimate in favor of the 20 MPa implied by earthquake stress drops [Brune and Allen, 1967]. As McKenzie admits, there is little justification for this choice. The amount of stress drop during a seismic rupture can only be a lower bound on the shear strength of the lithosphere. On the basis of his choice of 20 MPa a 100-km-thick elastic litho-

sphere is necessary to maintain the subduction zone gravity anomalies against shear failure. The longer-wavelength anomalies represented in the satellite gravity field and the geoid, however, require 100 MPa or more of stress even for a 100-km-thick lithosphere. McKenzie concludes that these anomalies must be maintained by flow in the mantle and may be a consequence of small temperature inhomogeneities.

Kaula [1969, 1972] continues with McKenzie's theme that the support for the satellite-derived gravity field must lie below the lithosphere and entails flow in the mantle. He proposes a tectonic classification of the long-wavelength gravity anomalies corresponding to degrees n -- 6 to n -- 16 based on an association with tectonic features such as subduction zones, ridge crests, orogenic belts, sedimentary basins, and areas of Pleistocene glaciation. Of the 11 classifications, 6 correspond to currently active tectonic features. If indeed the lithosphere is incapable of supporting statically the gravity anomalies over these features, then the magnitude and sign of the gravity anomalies provide information concerning mantle convection.

Lambeck [1972] challenges the assumption that the anomalies observed in the sateBRite gravity field cannot be supported by the lithosphere. His greatest objection is with McKenzie's 20-MPa stress limit, which he considers to be too low by at least a factor of 4. Lambeck also concludes that McKenzie's

stress estimates are overestimated by a factor of 2 due to an

error in McKenzie's equations and his use of a two-dimensional geometry.

For a three-dimensional geometry, Lambeck constructs a table of average maximum stress arising from gravity anomalies of degree n' and higher, assuming they are supported by a 100-km-thick lithosphere (Table 3). With Lambeck's preferred 100- to 150-MPa estimate for the critical strength of the lithosphere, it is possible statically to support anomalies of degree 8 or 9 and greater.

Lambeck [1972] emphasizes the point that the broad positive anomalies described by harmonic degrees n -- 8 and n = 9 over spreading centers need not be maintained by convective forces, but this conclusion depends on the assumed thickness of the lithosphere. Isostatic studies indicate that the mechanical lithosphere is only 30 km or so thick. For a more realistic lithospheric thickness, the stresses in Table 3 would increase by a factor of 9.

Chase [1979], like McKenzie [1967] and Lambeck [1972], favors the forward approach to modeling the long wavelengths of the gravity and geoid anomalies. Rather than considering the effects of a warped lithosphere, he attributes the anomalies to uncompensated point masses at depth. Much of the geoid character in harmonics 10 through 20 can be explained by positive mass anomalies in the worldwide subduction zone system. The stresses implied by the uncompensated mass range from a minimum 22 MPa for the Zagros subduction zone to a maximum 162 MPa for the Ryukuyu Trench, assuming a 100-km-thick plate. The required mass excess is, in most cases, less than the amount predicted by thermal plate models [e.g., McKenzie, 1969]. Thus stress estimates based on a mathematical formulation of density excess in a subducting slab will be larger than those consistent with the gravity field.

Although the models of McKenzie, Lambeck, and Chase are useful for estimating stress implied by the global gravity field, these models are static and thus do not answer the question of whether the lithosphere alone supports its anomalous mass and sustains its deformation. Nevertheless, the reoccur- ring theme from these studies is that for a variety of plausible anomaly sources, gravity anomalies with wavelengths between 5000 and 1000 km require that the earth in some way (con- vectively or otherwise) maintain stress differences from 20 MPa to near 200 MPa.

OCEANIC STUDIES

Long- Wavelength Anomalies

The vast improvement in satellite and surface-ship gravime- try fostered new attempts in the mid-1970's to determine the source of long-wavelength gravity anomalies [Anderson et al., 1973; Menard, 1973; Weissel and Hayes, 1974; $clater et al., 1975; Watts, 1976]. These investigations all center on oceanic observations in order to avoid the complexities of continental crustal structure and tectonic history, and they share a common line of reasoning. Numerical simulations of mantle convection in a NewtonJan fluid [e.g., McKenzie et al., 1974] predict a positive surface elevation and gravity anomaly over rising convection limbs for both high and low Rayleigh number flow. Thus positively correlated gravity and depth anomalies provide a strong case for convection in the mantle, partic- ularly when lithospheric sources for the anomalies can be ruled out. Although the calculation of residual depth anomalies is tedious in that it involves systematically correcting bathymetric data for sediment loading and the empirical age-


Free o•r Grovdy Anomaly

•=75km

Te:50 km .• -- Observed

+ 32_.5

+221

Te=30km

Te •O•m_. F Te=lOk

I00

+16.6

+198

+ 395

MGAL

Topography ,

IOO0] •. IOOKM 0 Meters 01 . .•!.'::, :37906

_1000• ?- ' ...

Fig. 2. Comparison of observed and computed free air gravity anomaly profiles of the Hawaiian ridge near Oahu. The computed profiles are based on the elastic plate model and assumed values of Te of 10, 20, 30, 50, and 75 km. The best overall fit to the observed data is for Te -- 30 km. From Watts et al. [1980].

depth relation [Sclater et aL, 1971], the remainder of the analysis consists of simply calculating regression lines of gravity on bathymetry.

Anderson et al. [1973], for example, claim that a 0.33 gravity unit (gu)/m correlation between gravity and bathymetry holds for the worldwide ocean-ridge system. On an ocean-by- ocean basis, however, the correlation is convincing only for the Atlantic and the Southwest Indian Ridge [see Anderson et aL, 1973, Figure 3], and even the Indian Ocean correlation disappears if the data from the Madagascar plateau are removed. In addition, Watts [1976] questions the reliability of the results of Anderson et al. [1973], Menard [1973], and Weis- sel and Hayes [1974], which are all based on the SE 2 gravity field [Gaposchkin and Lambeck, 1971]. A comparison of the SE 2 field with sea surface data and the more recent Gem 6

solution [Lerch et aL, 1974] reveals that anomaly peaks can be offset as much as 900 km, half the wavelength of interest, in the SE 2 field.

In any case it is unlikely that analysis of very long wavelength gravity and topographic anomalies will provide much information concerning the strength of the lithosphere. Re- gardless of whether or not the lithosphere could support the observed gravity anomalies, the agreement between theoretical calculations and observed regression slopes favors convective support. The case for convection could be made even

more convincing by comparing the ratio of Fourier transforms of the gravity and bathymetry data with the theoretical admittance from convection models:

Z(k) = C•)/H(k)

[McKenzie, 1977], where uppercase variables denote Fourier transforms of gravity g and topography h and k is the modulus of k. Most of the analyses computed a regression from 5 ø x 5 o data averages and therefore give an admittance estimate at only one k value.

In addition, from what is known of lithospheric behavior through isostatic compensation and glacial rebound studies, it is uncharacteristic of the lithosphere to sustain indefinitely uncompensated warps thousands of kilometers in extent without some incentive from below. Even if we do suppose that upward flexures are caused by the rising of low-density material from the mantle, the gravity anomalies are too ambiguous to define the distribution of the load. The magnitude of the deviatoric stresses associated with the observed surface strain

could be several tens of MPa or several hundred MPa, depending on how the support is applied.

Isostatic Compensation

In the oceans, about 50% of the power in the gravity spectrum is related to the compensation of oceanic features [Watts, 1978; Cochran, 1979; McNutt, 1979]. These gravity anomalies, in the wavelength range from 20 to 1000 km, are distinctly of lithospheric origin and thus pertain more directly to the problem of lithospheric strength. The choice between local and regional compensation for a particular feature can be decided on the basis of gravity anomalies alone. Figure 2, from Watts et al. [ 1980], compares the observed free air gravity anomaly over Oahu with theoretical gravity from elastic plate models. An extremely thin plate model (Te -- 10 km) poorly predicts the observed gravity; the anomaly from a local compensation mechanism (Te -- O) would give a worse fit yet. Large positive free air gravity anomalies flanked by encircling troughs over Oahu and other islands and seamounts demand regional compensation. Such features therefore are more likely to place a lower bound on the strength of the lithosphere; they are very large, relatively uneroded, and regionally supported.

Models proposed to explain the bathymetry, gravity anomalies, and deformation of the Moho boundary in the vicinity of these large islands and seamounts assume that the asthenosphere behaves, for loading time scales greater than 30,000 years, as an inviscid fluid and that the lithosphere can be described by one of the following theologies: (1) perfectly elastic [Gunn, 1943; Walcott, 1970b], (2) elastic with discontinuig. y at a free edge under the load [Walcott, 1970b; Watts and Cochran, 1974], (3) viscoelastic, (4) layered viscoelastic with viscosity decreasing with depth [Suyenaga, 1977], and (5) elastic- perfectly plastic [Liu and Kosloff, 1978]. The simplest model, a continuous elastic plate, almost certainly over-estimates the stress for a given load. Any attempt to incorporate more realistic theological behavior into the lithosphere can only serve to reduce the implied stress. For the purpose of this discussion we wish to determine how much the stress can be lowered

without violating the observations. The trend in recent years toward increasing complexity and

number of free parameters in the models has been dictated more by dissatisfaction with the implied stresses than by re-


Fig. 3. Examples of two elastic plate models. Upper model: continuous elastic plate of thickness T•; Pc is the density of the material overlying the lithosphere and Pm is the density of the asthenosphere. Lower model: elastic plate fractured under the load axis. From Walcott [1976].

fmement in the observations. Consider, for example, the thin- plate equation

Dvnw(x) + Apgw(x)= P

where D is the flexural rigidity given by (1), w the plate deflection, Ap the density contrast for materials overlying and underlying the plate, and P the applied load. The solution for plate deflection of a continuous elastic sheet (Figure 3) under a line load, such as a seamount chain, is

w = exp (-x/a)A[cos (x/a) + sin (x/a)]

in which the flexural parameter a is related to D by

Ol 4 '- 4D/Apg

The value of A is determined by the isostatic condition

and therefore

P = 2 Apgw(x) dx

A = P/2Apga

The bending stress ox is proportional to d2w/dx2. '

-2EzA

o,, = a2 exp (-x/a) [cos (x/a) - sin (x/a)]

where E is Young's modulus and z the vertical distance within the plate from the neutral plane. The maximum for this function occurs at x -- 0 and is given by

-EzP

Oxmax = Apga3 Walcott [1976] calculates that the maximum bending stress

at the base of a 60-km-thick plate loaded by a seamount chain

3 km high and 30 km wide is 200 MPa. Even taking into account the three-dimensionality and distributed nature of real loads, McNutt and Menard [1978] calculate 200-MPa stress under Tahiti using a best fitting 14-km elastic plate thickness. Walcott [1976] considers 200 MPa to be the 'crushing strength of rock' and predicts that failure should occur.

Model 2 in Figure 3 reduces the maximum stress by remov- ing the point of maximum curvature. The equation for the deflection of a plate with a free edge beneath the load is

w = B exp (-x/a) cos (x/a)

where

B = P/Apga

The bending stress is

-2EzB

o• = a2 exp (-x/a) sin (x/a)

and reaches its maximum at x/a = •r/2 (the first nodal point):

-2EzP

O .... -- Apa3 exp (-•r/2) A load P that would produce 100 MPa of stress on a continuous plate would produce only about 40 MPa bending stress on a fractured plate. While the discontinuous plate reduces the implied stress, Walcott [1970b] found that he was unable to si- multaneously fit Moho depth, flexural wavelength, and bathymetry with one value for the flexural rigidity of the fractured plate. Moreover, while it might be reasonable to suppose that during magmatic activity the lithospheric plate is weakened and possibly decoupled below the load, it is difficult to justify a free edge boundary condition for regions in which volcanic activity has ceased [Liu and Kosloff, 1978].


A viscoelastic plate can also reduce stress for a given flexure profile. The behavior of a plate with a viscous in addition to an elastic response is to relax its elastic stresses as loading time increases. Given an initial viscoelastic rigidity Do for loading times short in relation to the viscoelastic relaxation time r, the plate appears to be perfectly elastic. For loading times long in relation to r the flexural wavelength decreases, and the amplitude increases. If we were to interpret a viscoelastic plate profile in terms of the perfectly elastic equations, it would appear that the flexural rigidity was less than Do. The important point for the implied stresses is that the observed curvature of the plate is no longer proportional to the stress because some of the strain is nonrecoverable, nonelastic deformation caused by viscous flow. For any t > 0 then the stress will always be overestimated if the strain is assumed to be perfectly elastic. Nadai [1963] derives the following expression for the remaining elastic deformation w' in terms of the viscoelastic deflection w at a

time t after the emplacement of the load:

t w'(t) -- w(t) - exp (-t/r) w(t') exp (t'/r) dt' (2)

The reduction in maximum stress for a viscoelastic rheology as compared with an elastic one depends on the ratio t/r, and therefore we need an estimate of r for the oceanic lithosphere.

Walcott [1970a] interprets numerous apparent flexural rigidities D' from continental and oceanic loading studies in terms of a viscoelastic plate model. From a trend toward decreasing D' with increasing load age he estimates r -- 105 years. There are several problems complicating Walcott's interpretation. Known differences in continental and oceanic thermal structure suggest that apparent elastic thicknesses will vary. Firstly, some of the differences in D' may result from variations in initial rigidity Do quite apart from any relaxation. Secondly, the loading times for Pleistocene lakes Algon- quin and Agassiz are less than 10,000 years. The asthenosphere cannot be treated as a fluid on such short time scales, and the loads no doubt did not reach equilibrium. Finally, the relation between D' and Do for linearly viscoelastic plate is [McNutt and Parker, 1978]

where

D' -- Do exp [-t/r(1 + 14k4)] 1 + ink 4 {1 - exp [-t/r(1 +/4k4)]} (3)

P = Do/Apg

Apparent flexural rigidities can only be compared if they are determined from loads of the same wavelength • -- 2•r/k.

A value for r as low as 10 • years would indeed greatly decrease the elastic stresses implied by present-day loads. We can approximate the reduction in stress for a given flexural profile when interpreted in terms of a viscoelastic rather than elastic plate model by using (2) and the equation for viscoelastic deflection at time t from a constant line load [Nadai, 1963]:

w(x) -- 2-•pg' (e -yø cos Yo - e -y cos y)

1 [e_•o (cos Yo Yo)l [1 In (ao/a)l} (4) + -- + sin -

in which

a04= 4Do Apg Yo = X/ao y = x/a and

4t/r = (ao/a) 4- 1 - In (ao/a) 4

The integration of (4) is not easily accomplished with the system of equations in this form. For small t, however, we can estimate an upper bound for the remaining elastic deflection w' by putting a lower bound on $w exp (-t/r) dt:

w'(t) = w(t) - e-'/'w(O) e r/' dt'

= w(t) - w(O){1 - exp (-t/r)} (5)

Substituting for w from (4) into (5), we obtain

P { 1 (e_•o cos Yo - e -• cos y) 2-pg

I [e_yo (cos Yo - In (ao/a))] } + -- + Sin YO) ( e-t/' 0lo

The bending stress is proportional to the second derivative of w', which has a singularity at the origin. In the physical world, however, we would not encounter this point of infinite viscoelastic curvature, since point loads do not exist. For points other than x -- 0 the curvature remains finite and is

d•w ' p

dx • = Apgao 3 {yo-3(e -yo cos yo - e -y cos y)

+ yo-2[e -yø (cos Yo + sin Yo)- (ao/a)e -• (cos y + sin y)]

+ yo-'[e -yø sin Yo - (ao/a)2e -y sin y]

+ [e -'/' - In (ao/a)le -yø (sin Yo - cos Yo)}

This expression was evaluated at the crest of the first flexure arch for values of t/r = 0.02568 and t/, -- 3.0568, with ao -- 50 km. In each case the stress was compared with the stress implied by a purely elastic plate with the same flexural wavelength. For the short loading time t/r -- 0.02568 the reduction in stress for the viscoelastic rheology is 5%. When the loading time is of the order of 3 times the viscoelastic decay time, the stress reduction calculated is 15%, but this is only a lower bound, because for large t the approximation made in eval- uating the integral in (2) is no longer valid. Therefore we may conclude that interpreting flexure profiles in terms of a viscoelastic rather than an elastic plate model leads to lower stress estimates, but the difference is significant only for loads much older than r. If r is of the order of 100,000 years, we could presume that for loads a million years or older, appreciable stress relaxation has occurred.

However, flexural evidence from the Hawaiian-Emperor seamount chain is incompatible with a 10 • year value for r. Assuming an initial plate thickness of 90 km, Watts [1978] finds that the present-day 20- to 30-km apparent plate thickness under the island of Hawaii requires a r value of 105-106 years. Flexure beneath the older Emperor seamounts north of


0 Km 20 0

I 80

60 I00 140 I I I

o • ,ooo "•] r ø •ooo

b 0 Om • ' m • i I Deflection

Fig. 4. Creep in a finite element viscous-elastic plate under constant load. (a) Cross section of an 80-km-thick plate showing the spread of fluid elements as time progresses. The numbers indicate which regions of the plate have relaxed to the point of becoming fluid elements at 1, 3, and 5 x 10 n years after the load is applied. These regions support no shear stress. (b) Vertical displacement w at the plate surface. The arrows indicate the location of maximum displacement at 0 and at I and 3 x 10 n years. After Suyenaga [1977].

40øN implies Te -- 10-20 km and r -- 106-107 years. Watts interprets this discrepancy in viscoelastic relaxation times to mean that significant relaxation does not occur in the Pacific plate over 50-m.y. time scales. He prefers a model in which flexural rigidity does not decrease in time but is a function of the age and therefore of the thickness of the oceanic plate when the load is applied. The lower flexural rigidities for the Emperor seamounts relative to the Hawaiian Archipelago are explained by a younger age, and therefore a lower elastic thickness, for the region of the Pacific plate beneath the Em- peror Seamounts when the chain was created 50-60 m.y. ago. The very low flexural rigidity and thin elastic plate thickness found by Cochran [1979] and McNutt [1979] for the topography in the vicinity of Pacific spreading centers supports Watts' interpretation.

Regardless of whether or not viscoelastic relaxation does occur, the implications for stress are not affected for long decay times. For r greater than 50 m.y., no appreciable stress relaxation has occurred for Hawaii, Tahiti, and other relatively young oceanic loads. Both elastic and viscoelastic plate models predict stresses exceeding 100 MPa.

Another variation on the elastic plate model is the layered plate in which relaxation begins at the base of the lithosphere and migrates upward. This scheme is appealing from a theological viewpoint because the temperature dependence of viscosity predicts that viscosity decreases with depth. Suyenaga [1977] uses a finite element scheme [Zienkiewicz, 1971] to in- vestigate the time history of flexure for a plate with depth- dependent viscosity. He terms this plate model 'viscous- elastic' to distinguish it from the viscoelastic model whose properties are constant with depth. Suyenaga's procedure can be outlined as follows:

1. Apply a load at time zero to a plate 100 km thick, and using the finite element method, numerically calculate the elastic displacement and stress according to the elastic plate equations.

2. Calculate the amount of creep that will occur at any point within the plate during a time increment At according to a creep law based on experimental data [Weertman, 1970]

• = •1o n exp (-GTm/T) (6)

in which

• strain rate; •1 = 3.4 x 108 kbar -n s--l; n=3; G-- 30;

Tm melting temperature at ambient pressure; T actual temperature at depth z; o deviatoric stress.

The numerical constants in this empirical creep law are taken from the work of Carter [1975], Weertman and Weertman [1975], and Kirby [1977]. The creep strain is estimated from the elastic stresses and homologous temperature (T/Tm) curve from Mercier and Carter [ 1975].

3. Assume the creep strain is an initial strain within the plate and return to step 1. When the creep strain in a plate element becomes greater than the elastic strain, the element is considered to be a fluid.

Elastic elements begin to become fluid at the base of the lithosphere first where T/Tm is a maximum, although, as shown in Figure 4a, the pattern is also influenced by the stress distribution from the flexure. The flexural rigidity of the plate therefore decreases with time because the nonfluid portion of the thermal (• 100 km) lithosphere that actually supports the load decreases with time. The upward spread of fluid elements practically halts after about 106 years at depths between 15 and 30 km, where T = 0.3Tm to 0.5Tm [Murrell, 1976]. Below this depth interval, stress relaxation is complete, while above, relaxation is only partial and does not change appreciably over loading time scales. As a result of the partial relaxation the stress in the upper, long-term elastic portion of the plate is less than what would be estimated from elastic theory. The evolution of the surface displacement as the fluid elements mi- grate upward, shown in Figure 4b, is similar to that of the viscoelastic lithosphere with no depth dependence in the viscosity. The important difference in the two models is that relaxation eventually ceases in a viscous-elastic lithosphere before the effective rigidity reaches zero.

The viscous-elastic model is intended to explain why the apparent elastic thickness of the oceanic lithosphere is only a fraction of its seismic thickness [Hanks, 1971, 1977]. We can also estimate its long-term stress implications by considering the strain rate at the base of the 30-km-thick elastic layer assuming T = 0.5 Tin. From (6), • -- 2.4 X 10-m7/S, which equals 7.5 x 10-n/m.y., for a deviatoric stress of 200 MPa. In the viscoelastic plate model it is assumed that strain rate depends linearly on stress:

• = (1/•t)o •t = Er/3

•_•o

• øo 5 I0 15 STRAIN, • (0/0)

Fig. 5. Stress-strain curve for dunite at 500-MPa confining pressure, 800øC, and 5 x 10 -4 s -• strain rate. After Griggs et al. [1960].

6386

Prediction of the Rheology of the Flexed , ,

from Olivine Deformation Maps

Lithosphere

Depth(km.) for older oceanic lithosphere 0 20 40 60 80 i00

[ • I ....... [ ..... I ...... • ...... I .. [ ,._.1 .........

:;' / ,7'• '%, '• • - •

i . •, • /N?, ¾, xx . •-- .', -- . o Moxirnurn / ,•-•,• X • •.Xl • xxx • n d 10 •, bending / '., 'VI•% •i • '• ,I • x •- I0 '1o , - io2 I/_. ? / X ,¾|lJ•'\x\ ,E •, _'•J r-•Jvvr_.• L_/-WV

/ t, CREEP / ' /!.' "• \ ..-, ! / ;/ / / qj\ ,"P, _.. ,

J/ CREEP dJ=lV '._ - 2" / . J,".;,-r18 •\,,-r16 i• Ira,-, -•_ • • --- I0 /I - ø

,,

j/, .. I "Elastic ' ! Temperature (øC x l0 )

C'i!!!i/ ' ' ' ' ' /•. C•?•.•.,•c Diffusional ..... ( Newtonian )

/, Cree•' • / /// ////.

[

I

•;• Plastic Law : •:•:• ....................................... Base of the Lithosphere

Rhealogical Zonation of the Lithosphere Fig. 6. Olivine deformation map with superimposed upper bound stress distribution in the lithosphere below a point

load. The deformation map is for olivine with a 0.1 grain size and includes the effect of increased confining pressure with depth. The fields are labeled according to the steady state rheology that dominates at that stress and temperature. Also included are the predicted strain rates (dashed lines); therefore effective viscosities can be calculated. The base of the lithosphere (thickness 100 km) is defined by the temperature of 1300øC. For a thinner lithosphere the depth scale can be read as 'percent of lithospheric thickness.' The lithospheric stress profile plots magnitude of stress versus depth. Stress in this context may be interpreted as bending stress, shear stress intensity, or deviatoric stress, all of which have very similar values. The lower part of the figure shows a projection of the fields crossed by the stress distribution onto a section of the lithosphere and is labeled according to the dominant rheology predicted for that zone. 'Elastic' implies that the strain rate for that zone will be sufficiently slow that it will appear elastic over long time scales. From Ashby and Verrall [1977], as modified by Beaumont [1979]. Reprinted with permission of Beaumont [1979] and Elsevier Scientific Publishing Company.

A strain rate similar to the viscous-elastic rate would occur in lithosphere supports the load. The increase in stress as the the viscoelastic plate for •7 -- 8 x 1024 N s/m 2 or •- -• 10 m.y. plate thins also increases the strain rate, but the factor T/T,. Thus at least initially, the relaxation rate at 30-km depth in a dominates the behavior. For example, to obtain a strain rate viscous-elastic plate is quantitatively similar to the rate of re- as high as 7.5 x 10-4/m.y. at a depth of 15 km where T -• laxation in a viscoelastic plate with •- = 10 m.y., assuming the 0.3 T,,,, the stress would have to increase to 4,400,000 MPa. same stress level. A major difference in the two models is that In general, the ultimate, long-term stress on time scales of while viscoclastic relaxation proceeds throughout the depth of several hundred million years for a viscoclastic plate model the plate, in the viscous-elastic model, relaxation at deep lev- would be of the order of the weight of the load (several tens of els increases the stresses in the upper levels, since less of the megapascals), since the compensation would approach that of


Airy. For a similar time scale the stress in the upper 15 km of a viscous-elastic plate would continue to increase in the range of several hundred megapascals, since the low T/T,, ratio would preclude relaxation at shallow depths. In this respect then the stress estimate is quite similar to that of the purely elastic plate. Consideration of laboratory creep data has merely provided a mechanism to thin the seismic lithosphere from 100 to the observed 30 km on a million-year time scale. It is perhaps significant to point out that if we were to interpret the change in flexural profile in a viscous-elastic plate in terms of a viscoelastic plate equation, it would appear that ? increases with time. This is exactly the behavior noted by Watts [1978] along the Hawaiian-Emperor chain.

The elastic-plastic lithosphere [Liu and Kosloff, 1978] is a further refinement of plate mechanisms incorporating rock deformation data. The stress-strain relation for an elastic-perfectly plastic material is shown in Figure $. The elastic behavior is represented by the sloped line along which stress and strain are linearly proportional. At a critical stress level Oc determined by the rate at which the load deforms the lithosphere, the material behavior is plastic. Strain increases without change in stress. The strain rate dependence of the yield stress is given by a power law equation extrapolating laboratory data to geologic loading rates [Carter, 1976]:

• = AOc n exp (-Q/R2) (7)

in which Q is the activation energy and R the gas constant. The ultimate strength of the lithosphere is equal to Oc and depends on temperature (and thus depth) as well as strain rate. For lower strain rates corresponding to geologic loading, the yield stress also is lower.

Liu and Kosloff [1978] model flexure under the Hawaiian archipelago using an elastic-plastic lithosphere. The rate dependence of the elastic response is incorporated by using 90% of the elastic moduli, based on a 2% decrease for each 3-dec- ade decrease in loading rate. The plastic part of the deformation is given by (7) using the parameters from Carter and Ave Lallement [1970]. Liu and Kosloff successfully fit a profile northeast of the island of Oahu, but there are many free parameters in the model, and the fit is in no way unique. Since stress caused by any load cannot exceed Oc, the stress implications for the elastic-plastic plate are determined by the assumed temperature structure and deformation rate in (7). For the parameters selected by Liu and Kosloff, strength varies greatly with depth, and the nonyielded portion of the lithosphere is about 30 km thick. This study is meant to illustrate how rock deformation data can be employed in flexure studies. For the Hawaiian ridge, at least, there appears to be no conflict between the extrapolated results of deformation experiments and observed lithospheric flexure from large long- term geologic loads.

To summarize this section, we find that the proposed rheological models can explain the regional compensation for features like Hawaii as long as they predict a 30-km-thick, pre- dominantly elastic layer that sustains 100 MPa or more stress on a time scale from 1 to 50 m.y. Model 1, the homogeneous elastic plate, is merely the simplest mechanism that satisfies this requirement. The fact that a one-parameter elastic model can explain the data testifies more to limitations of the observations than to the rheological simplicity of the earth. Rock strength depends on a number of parameters, including rock type, confining pressure, temperature, and strain rate. Accord- ing to a review by Kirby [1980] the effective mechanical thick-

ness of the lithosphere will be reduced from above by brittle fracture and from below by ductile deformation. Strength decreases with depth according to the temperature structure, leaving an elastic core several tens of kilometers thick with a yield strength between 300 and 800 MPa.

Beaurnont [1979] demonstrates the theological complexity in a flexed lithosphere by superimposing the deviatoric stress predicted by a 100-kin-thick elastic plate on Ashby and Ver- rall's [1977] deformation map for olivine (Figure 6). His model only applies to the instant in time before creep and plastic failure redistribute the stress, but in a general way it in- dicates the type of response expected. If preexisting zones of weakness occur in the upper 10 to 15 km of the plate, the region yields by brittle failure on faults. Even in the absence of faults the lithosphere can deform by cataclastic flow involving stable microfracturing at depths between 2 and 20 km [Kirby, 1980]. The lower 20 km of the lithosphere yields plastically. Rapid relaxation by diffusion and power law creep at 50- to 80-km depths also reduces the effective elastic thickness of the lithosphere. The central part of the plate between 15 and approximately 50 km relaxes so slowly that for million-year time scales it appears elastic.

To only a first approximation therefore can we model the mechanical lithosphere as an elastic plate 30-40 km thick. The more complete picture from the rock mechanics literature predicts that with detailed observations we should also detect (1) a time dependence in the flexural rigidity as diffusional creep gradually thins the elastic plate and (2) a stress dependence in the flexural rigidity as the volume of plastically yielded lithosphere increases with deviatoric stress. The time-dependent aspects of flexure are not evident over the loading times (~200 m.y.) observed in the oceans. If relaxation in the elastic core does occur, the time scale for the creep is greater than 50 m.y. For oceanic islands and seamounts, Beaumont [1979] finds that the limited zone of plastic failure changes the surface displacement by 'only 10%. In the absence of reliable data on Moho displacement, deviatoric stresses associated with topographic loading may not be large enough to resolve details of plastic deformation using surface observations alone.

The Outer Rise

The case for plastic yielding is far better documented in the lithospheric flexure profiles seaward of subduction zones. Here the strain reaches about 2%, compared with strains of less than 1% for seamount loads [Watts et al., 1980]. In the region of the outer rise, approximately 100 km from the trench axis, the lithospheric deformation is adequately described by the elastic plate model. The effective elastic thickness in the range 30-40 km [Hanks, 1971; Watts and Talwani, 1974] agrees well with the results from seamount loading studies. Nearer to the subduction zone, on the outer trench slope, the extremely large curvature in the plate requires plastic yielding. McAdoo et al. [1978] estimate 470-720 MPa for the yield stress. Similar results are obtained in other studies [ Turcotte et al., 1978; Bodine and Watts, 1979].

Not all investigators agree that the outer rise is supported by the strength of the lithosphere. Melosh [1978] proposes an alternative scheme in which the outer rise is produced by a momentum change in the flow of a viscous layer beneath an elastic lithosphere (Figure 7a). Since it is not required that the outer rise result from forces and moments applied at the trench axis, the wavelength of the deformation does not determine the thickness of the elastic lithosphere. Deviatoric


SPERE

Fig. 7. (a) Dynamic model for outer risc topography. A thin elastic plate passively rides over a moving viscous layer. (b) Variation of dynamic model in which the elastic plate is faulted, forming independent blocks.

stresses can be made arbitrarily small by thinning the elastic plate. For the old oceanic lithosphere common to the outer rise region, however, it would be unreasonable to suppose that its mechanical thickness is less than the 30- to 40-km thickness beneath seamounts. Regardless of the dynamics of formation for the outer rise, the observed strain still implies several hundred megapascals of deviatoric stress in a 30- to 40-km-thick plate.

A variation of the dynamic support model assumes that the upper lithosphere consists of faulted blocks independently rid- ing the deeper viscous flow, as shown in Figure 7b. This model assumes that the lithosphere is devoid of shear strength. Seismic refraction profiles (see Watts et al. [1980] for an example) show that normal faults are indeed a common feature of outer trench walls. Brittle behavior in the upper 10

CONTINENTAL STUDIES

There is little doubt that oceanic topography formed on a cooled, thickened lithosphere is regionally compensated with associated high stresses (hundreds of megapascals). Can the same model be applied to the continents, and, if so, what is the rheology of the continental lithosphere over billion-year time scales? We can anticipate several problems in the analysis of continental compensation:

Nature of the load. Implicit in most loading studies is the assumption that the lithosphere passively responds to a load applied from above. A good case for this simplification can be made for sedimentary basins with distant sources. While for volcanic loads the assumption is less justified, the plate is al- tered and weakened by magmatic activity in the area least re-

km of the oceanic plate under large tensile stress is predicted solved by the data. Only when complete refraction data under by the rock deformation experiments [Kirby, 1980]. In fact, if volcanic loads become available will it be necessary to deal re- evidence of surface fracturing were lacking, there would be alistically with the load-forming process. The loading assump- good reason to doubt whether laboratory data do apply to geologic processes. At this time, however, there does not appear to be any basis for extending the faults 40 km to the base of the elastic plate, and the model can be rejected for both rheological [Murrell, 1976; Ashby and Verrall, 1977; Kirby, 1980] and seismic reasons [Hanks, 1979].

tion is least appropriate for continental mountain belts. The details of orogeny are sketchy, but in most cases, topography is created as lithospheric plates collide. The distinction between the 'load' and the lithosphere becomes meaningless if the entire thickness of the lithosphere is deformed.

Complexity of geologic history. Overprinting of several


orogenic, intrusive, and extrusive volcanic events is common for older continental regions but rare in the oceans. Ever present erosion in subaerial environments introduces a time de-

pendence to the loading history.

Sedimentary Basins

While the sedimentary basins involve loading conditions which best fit the idealization in the theoretical model, the theological implications from various studies show no con- sensus. For example, Haxby et al. [1976] infer from the configuration of successive sedimentary facies that the effective rigidity in the Michigan Basin has increased to 4 x 1023 N m over a time span of approximately 100 m.y. They invoke uplift of the gabbro-eclogite phase transition caused by shoaling isotherms as the driving mechanism for gravitational subsidence of the basin. An increase in the flexural rigidity would be an expected consequence of cooling and thickening a purely elastic lithosphere. Nunn and Sleep [1979] successfully model the same basin as loading on either an elastic plate with D -- 2 x 102'N m or a viscoelastic plate with Do -- 10 •4 N m and • = I m.y. Thermal contraction is used as the driving mechanism for subsidence. For the North Sea, Beaumont [1978] prefers the viscoelastic crustal model to explain the apparent decrease in flexural rigidity with time. In a manner expected from a viscoelastic theology the subsiding part of the basin narrows with time, and erosion of the peripheral uplift regions results in an apparently regressive stratigraphic sequence at the basin surface. This pattern of progressively younger sediments outcropping toward the center of the basin is typical of many basins. Sclater and Christie [1980] explain the observed crustal thickness, heat flow, and subsidence of the North Sea by invoking 50-100% stretching of the Central Graben. They deliberately ignore flexural loading of the lithosphere on the assumption that thermal subsidence and faulting are the dominant factors controlling the evolution in the center of the basin. Clearly, the interpretation of sedimentary facies in terms of a theological model is nonunique, with the mechanism of subsidence contributing the most uncertainty. Studies of the structure and evolution of sedimentary basins and their one-sided analogues, passive continental margins, will not allow unambiguous estimates of rheological parameters until other factors are better determ'med. These factors in- dude the initiating mechanism, the sedimentary budget, and sea level history [Beaumont, 1979].

Distributed Topography

Individual volcanic features such as the island of Hawaii

are rare on the continents. Elevated regions occur in broad mountain belts where deformation from any one load is in- distinguishable from that of another. A useful method for analyzing the isostatic compensation in regions of complex topography is the response function technique. On the assumption that isostatic compensation for a point load is linear and isotropic, the isostatic response function Q(k) can be calculated from Fourier transforms of topographic and Bou- guer anomaly data [Dorman and Lewis, 1970]:

C•) -- Q(k). H(k)

where G and H are gravity and topography transforms and k = Ikl - (k• • + ky•) -'/•. The response Q calculated from actual data sets can be directly compared to theoretical 0 from the linearized forms of regional and local compensation models. For example, for linear Airy isostasy,

O(k) -- -2•rpG exp (-kz½) (8)

where p is the density of the topography, zc the compensation depth, and G Newton's gravitational constant. For the elastic plate compensation model,

{ k4O• -' Q(k) -- -2•rpG I + apgJ exp (-kz½) (9) where Ap is the density contrast for materials overlying and underlying the plate. Figure 8 shows theoretical isostatic response functions derived from (9) for two z½ values. The D = 0 curves correspond to local compensation. For identical z, values the elastic plate response (D = 1022 N m) has a sharper curvature an d falls off more quickly to zero at short wavelengths. The explanation for this behavior is that narrow features, supported by the strength of the plate, have smaller Bouguer anomalies and therefore lower Q values than similar features that are locally compensated. Broad features do not feel the effect of the elastic plate, and therefore local and regional responses are similar at long wavelengths. Because compensation depth varies, curvature, rather than amplitude of the observed Q, is the best estimate of elastic stresses. For example, referring to Figure 8, the local compensation (D -- 0) response at medium to long wavelengths with a 40-kin compensation depth is actually lower in amplitude than the D -- 1022 N m response with shallower (30 km) compensation.

A close relative of the isostatic response function is the free air response Z:

FA(k) -- Z(k). H(k)

where FA is the Fourier transform of the free air gravity anomaly. The Z response is most commonly used for oceanic surveys. The theoretical response 2 is related to 0 via

• = {) + 2•rpG exp (-kz,)

where z, is the average depth from the sea surface to the topographic surface.

There are two major limitations in using the response function technique. The first problem is that all gravity signal is assumed to be related to compensation of topography with constant density. Density variations unrelated to topography are a source of noise, but their effect can be minimized by sta- tistically separating out only the gravity field correlated with the topography. In regions of low topographic signal, how-

-12

-10

-O8

-0.6

-O4

-O2

o 5000

- 2 N rn '•x•L -- DD !01, , , ' - 0

2000 I000 500 200 100 50

X,km

Fig. 8. Theoretical isostatic response functions for local compensation (solid curves) and regional compensation with D = 10 •2 N m (dashed curves). Each response is calculated assuming two different values of compensation depth z½.


-I.4

-I.2 --

-I.0 --

-0.8 --

-0.6 --

-0.4 --

-0.2 --

0 --

0.2 5000

AUSTRALIA McNutt & Perker , 1978

U.S.A. Lewis & Dotman, 1970

CANADA Stephenson & Beaumont • 1979

2000 I000 500 200 I00 50

WAVELENGTH X , KM

Fig. 9. Comparison of United Statc• and Australian isostatic response functions with the response from the Canadian Shield. Bars indicate the standard error in each spectral estimate.

ever, the uncertainty in Q is large. A more serious problem in- v01ves the necessarily large dimensions of the region under in- vestigation. For the continents the characteristic falloff region for Q lies between 2000-km and 500-km wavelengths. The calculated Q may therefore give only an average rigidity estimate from several tectonic provinces within the survey boundary.

From the curvature in Lewis and Dorman's [1970] response function for the United States (Figure 9), Banks et al. [1977] determine that the apparent flexural rigidity in the United States is between 102• and 1022 N m. They conclude that Wal- cott [1970a] found higher rigidity values (1023-10 24 N m) from other North American features because he measured the deformation at the surface rather than at the Moro. Thus Banks

et al. [1977] imply that some sort of depth-dependent relaxation reconciles the two rigidity estimates.

McNutt and Parker [1978] find an even lower value for apparent flexural rigidity in Australia and invoke a viscoelastic relaxation model to explain the difference in United States and Australian D values. Assuming a 200-m.y. age for the latest Australian orogeny and a 50-m.y. age for the Laramide orogeny in the United States, the viscoelastic relaxation time •- = 45 m.y. If the viscoelastic theory is correct, the isostatic response from eastern Canada should resemble that of Australia because the region consists of Precambrian shield and orogenic zones no younger than the Paleozoic. Figure 9 compares the isostatic response from the Canadian Shield [Stephenson and Beaumont, 1979] with the Q estimates from the United States and Australia. The viscoelastic model fails this test; the Canadian response in no way resembles that of Australia.

The validity of McNutt and Parker's [1978] viscoelastic ,model hinges on the assumption that the response function is dominated by the signal from the most recent orogeny. This assumption is justified for Australia. When the continent is divided into eastern, central, and western grids, the most coher- ent Q which dominates the average Q is that of the geologi- cally youngest eastern section. In a similar manner the United States data set can be divided into two subsets: a western grid containing the Rocky Mountains, the Basin and Range, and the Sierra Nevada (Figure 10); and an eastern grid containing the Appalachian Mountains, the Great Lakes region, and the

Mississippi Valley (Figure 11). The response functions from the two grids are compared in Figure 12. The isostatic compensation is obviously different in the two regions, and Lewis and Dorman's [1970] response function was a hybrid of the two. The overall response reflects the response signal from the region containing the highest correlation between the gravity and the topography.

The Q functions in Figure 12 can be directly inverted for parameters in compensation models. We will consider the data in terms of the best fitting flexural rigidity D and compensation depth zc in an elastic plate mechanism. This model is appropriate for two reasons: the uncertainty in the data does not permit inversion for more than two parameters, and the results can be interpreted quite generally. If the compensation is actually local rather than regional, the preferred D value will be extremely low. If viscoelastic relaxation occurs, the resulting D can be interpreted as a time-dependent apparent rigidity.

The inversion results are presented in Figure 13. The horizontal scale plots flexural rigidity, and the vertical scale represents the one-norm misfit between the observed response and the theoretical model. Each curve corresponds to a different zc value. The dashed lines indicate the probability level at which the model is consistent with the data. For the western section

the best fitting compensation depth is between 30 and 35 kin, and the response rules out flexural rigidities above 10 •9 N m. Since this D value corresponds to a plate thickness of only 1 km, we conclude that compensation in the western United States is local. For the eastern grid, regardless of what compensation depth we assume, the best fitting D is 5 X 10 •2 N m, which implies a 20-kin-thick plate. The misfit valley is extremely sharp and strongly rules out local compensation.

Residual gravity maps give an indication of which features determine the isostatic response. The residual gravity map is produced by subtracting from the observed Bouguer gravity the anomaly predicted by filtering the topography with the observed response function. The residuals are 'isostatic anomalies,' although no particular compensation mechanism has been assumed. The residual gravity map for the eastern grid (Figure 11) shows a prominent positive anomaly over the mid-


WESTERN UNITED STATES ISOSTATIC ANOMALY

'"'• ...... >100 • WITHIN +_ IO0 g.u. :.'".....'• <-I00

200-g.u. CONTOURS

500 K M I

Fig. 10. Residual gravity anomalies in the western United States. Anomalies are the Fourier transform of G•k) - Q(k)H(k), where G and H are the wave number domain representations of Bouguer gravity and topography maps, respectively.

crust has lost whatever vertical mobility it possessed during orogeny. The stress implied by the gravity anomaly is 50 MPa, assuming that it originates at about 50-km depth.

In the western grid there are few surprises (Figure 10). The Rocky Mountains display anomalies consistently positive in sign, indicating that the compensation depth is generally deeper than the 30- to 35-km grid average. The Basin and Range shows negative residuals, indicating that the Moho is slightly shallower than the regional average. Both observations are consistent with the notions that the Colorado Plateau

has a root and the Basin and Range has an anomalously thin crust.

These results suggest that the tectonic process that formed the topography in the western grid, by some perhaps thermal or mechanical means, left the crust and upper mantle incapable of transmitting stress laterally. In the eastern United States, at least part of the lithosphere is capable of responding elastically. A very simple-minded model which could explain these results is shown in Figure 14. Perhaps the thin-skinned tectonics responsible for the Appalachian Mountains left the lower part of the elastic lithosphere essentially undisturbed. We would expect to observe an anomalously shallow compensation depth because the Bouguer gravity signal comes from the flexure of two interfaces with density discontinuities: one at the base of the deformed sedimentary layer and the other at the Moho located somewhere within the elastic layer. This type of compensation would result from shallow, compressional tectonics. In the western United States the topographic elevation is caused by tensional and/or vertical stresses affecting the entire crust and upper mantle. Whether differential movement occurred along faults, as shown in the schematic diagram, or high temperatures caused the rocks to behave viscously rather than elastically is immaterial for the purpose of explaining the gravity and the compensation. Firm conclusions require that individual loads be modeled; at this point we cannot rule out the possibility that several isostatic mechanisms, supporting features with different characteristic wavelengths, have combined to produce a meaningless response function.

Other studies support this conclusion that continental crust can sustain elastic stresses, given the right conditions of load emplacement. Both the Boothia Uplift [Walcott, 1970a] and

continent gravity high and anomalous lows in the Mississippi the midcontinent gravity high [Cohen and Meyer, 1966] ap- Valley. These gravity anomalies arise from subsurface density pear to be regionally compensated with a flexural rigidity of variations uncorrelated with the topography and therefore 2 x 1023 N m. The associated plate thickness, 30 km, is 10 km cannot be analyzed by the response function approach. The greater than the value determined for the Appalachian Moun- residual lows in the Hudson Bay area show the effect of still tains. It is difficult to explain the reduction in plate thickness incomplete isostatic rebound from the Pleistocene glaciers. beneath the Appalachian Mountains in terms of viscous relax- For the purposes of this study, the most important observation ation of elastic stresses. The ages of both the midcontinent is that the Bouguer gravity signal from the Appalachian gravity high (•1100 m.y.) and the Boothia Uplift (-•500 m.y.) Mountains has been essentially removed. The isostatic re- are greater than the age of the Appalachian orogeny (-•200 sponse in the eastern grid is determined by the gravity signal m.y.). From the model of the Appalachian Mountains pre- from these mountains. A residual belt of negative isostatic sented here, it seems more likely that the elastic lithosphere anomalies parallels the trend of the highest peaks in the Ap- was either mechanically thinned from above or partially re- palachians. This gravity low coincides with a 5•-•km-deep laxed at the base by elevated temperatures at the time of crustal root determined from seismic refraction data [James et orogenesis. al., 1968]. Given the present elevation of the mountains, the We face a dilemma in converting the continental com- size of the root is excessive and overcompensates the topogra- pensation models to maximum stress estimates: 200-m.y.-old phy. Thus the negative gravity anomaly results from crustal loads are extremely eroded. The present stress needed in the structure unrelated to present-day isostatic balance, although lithosphere to support the mountains will underestimate the it is entirely possible that the root is a remnant from a period peak levels attained in the past. To minimize the mitigating when the Appalachians were more impressive features. If this effects of erosion, we should concentrate on extremely young is the case, the persistence of the root may indicate that the loads, such as the Himalaya. Unfortunately, not only are the


EASTERN UNITED STATES ISOSTATIC ANOMALY

........ '"":'• > I00

• WITHIN __. I00 g.u. :".:• < -I00

200--g.u. CONTOURS o [ ,

500 KM I

Fig. 11. Residual gravity anomalies in the eastern United States.

necessary data unavailable, but the very fact that the moun- talias are young implies that some of their support may be dynamic. From the similarity in elastic plate thickness in the continents and oceans, we suspect that the continental lithosphere can also support deviatoric stress in excess of 100 MPa in areas where it has not been excessively thinned, heated, and fractured. Kirby [1980] suggests, however, that the weakening effects of water may play a more significant role in the continental granites than in the oceanic basalts.

EXTRATERRESTRIAL STUDIES

Many of the same techniques used for analyzing load defor-

emplacement of the oldest geologic units. These bodies are considered to be 'one-plate planets' [Solomon, 1978] and lack the complicated deformation patterns associated with the creation and destruction of lithospheric plates.

Solomon [1977] infers the thermal history of the moon and Mercury from the surface deformation. Early in a planet's thermal history the interior warms as core segregation progresses. The planet expands, causing tensional surface features and volcanism. When differentiation is essentially complete, the planet cools and contracts, producing compressional surface features and cutting off volcanic activity. For example, Mars is dominated by extensional features, indicating plan-

mation and determining stress on the earth have been applied etary expansion over most of its history. The absence of exten- to the moon and other planets in the solar system. The two most intensely studied from a stress viewpoint are the moon and Mars. While the remoteness of these planetary bodies makes it more difficult to interpret the geology and obtain precise measurements of gravity and topography, in at least one respect the analysis is simplified. There is no evidence from the moon, Mars, or even Mercury that there has been any horizontal shifting of rigid lithospheric blocks since the

sional features on Mercury places the planetary differentiation phase before the formation of the oldest surface features. The moon lacks extensive compressional or extensional features, and therefore its radius cannot have increased

or decreased by more than 1 km in the past 3.8 b.y. The 1-km limit is based on the assumption that a change in tangential stress of 100 MPa would be sufficient to produce observable surface faulting. Solomon [1977] finds that acceptable temper-


ature profiles T(z, t) predicting a maximum radius change of less than 1 km over the past 3.8 b.y. have initial profiles with melting temperatures down to 200- to 300-km depth and cold temperatures in the deep interior. However, the accumulated thermal stress in the lithosphere from the acceptable models is as high as several thousand megapascals. In the moon's interior the thermal stresses dissipate by flow on short time scales, but the mechanism and time scale for relaxation of the litho-

spheric thermal stress is unknown. The solution to this problem may have important implications for the state of stress in the lithosphere and the thermal evolution of planets.

Solomon and Head [1979] combine the model for global stress on the moon with predicted stresses from lithospheric flexure in order to explain the spatial and temporal distribution of linear rilles and mare ridges. From features 3.6-3.8 b.y. old they infer a relatively thin plate, 25-50 km thick. Younger features only 3-3.4 b.y. old require a 100-km-thick plate. Note that these effective plate thicknesses are several times those found on earth and imply that the long-term elastic layer is much thicker on the moon. Given the dependence of creep rate on temperature (equation (10)), a thick elastic layer may be the consequence of a more gradually sloping geotherm for the moon relative to the earth. The observation that the com-

pensation for younger lunar features requires a thicker lithosphere might also be consistent with a viscoelastic plate mechanism, but in this case a viscoelastic explanation is unlikely. Unless the initial plate thickness at time zero is much greater than 100 km, it is difficult for loads more than 3 b.y. old but only 0.1-0.8 b.y. apart in age to differ in effective plate thickness by 50%. From gravity and topographic data, Thurber and

Solomon [1978] find that the older lunar highlands are locally compensated, while the younger mascon maria are supported by the strength of the lithosphere. it follows that the lunar lithosphere must have increased in thickness after the formation of the highland topography.

Olympus Mons, a shield volcano sitgated on the Tharsis ridge of Mars, has received considerable attention recently. With a basal area of 3 x 105 km and a height of 25 km, Olym- pus Mons is the largest known volcanic feature in the solar system [Thurber and Toksoz, 1978]. The excess mass of Olym- pus Mons and the other three shield volcanoes of the Tharsis dome is sufficiently large to cause a 1.2-km anomaly in the gravity equipotential surface and account for about 6% of the planetary oblateness J2 [Reasenberg, 1977; Kaula, 1979]. De- bate has centered on whether or not the Tharsis dome volca-

noes are compensated. Phillips and Saunders [1975] conclude that while older regions of Mars are locally compensated, the topography of the Tharsis region is young and mostly uncompensated. Their observation could be explained by the scenario invoked by Solomon and Head [1979], in which the lithosphere thickens between the time of formation of young and old topography.

Thurber and Toksoz [1978] directly model the compensation of Olympus Mons using flexure theory. For an elastic plate thickness as small as 100 km, the model predicts 500 MPa for the maximum extensional surface stress. Thurber and Toksoz

see no evidence of surface faulting in response to the high stress, nor do they observe an arch of the order of 4 km high that would be produced by plate flexure. Their preferred model, an elastic plate approximately 200 km thick, predicts

WAVELENGTH (Kt4)

WESTERN u.s.

I I I I I I I

0 EASTERN U.S. & CANADA

- I i i

- 2.0 -I.6

(KI4 -I ) -3.2 -2.8 -2.4

LOG WAVE NUMBER K

Fig. 12. Isostatic response functions from the eastern and western United States.


• I i I I I WESTERN U.S. _

z• km._•__•_.

25

ompensation d

20

15

__zo_ yo___

. 50 %

t 35 km., • . . 66 % I0 -

/ I I I I I I I I

i016 i018 10 20 i0 22 i024 FLEXURAL RIGIDITY (Nm)

I I I I I I

EASTERN U.S. 8• CANADA

Compensation depth in km

25 km

30 km

35 km

40 km

45 km

1.5 %

20 %

5O %

80 %

I I I I I I I I I

i016 1018 1020 1022 1024 FLEXURAL RIGIDITY ( Nm )

25

2O

Fig. 13. One-norm misfit between observed response functions in Figure 14 and Q from elastic plate models. Plate parameters are flexural rigidity D and compensation depth Zc. Dashed lines indicate the probability that the observed misfit is caused by random errors in the data.

an arch amplitude of only a kilometer or two, in agreement with the topographic data, and produces maximum surface stresses less than 100 MPa. At depth, however, the maximum stress under the load would be greater than 100 MPa. Using Jeffreys' [1924] calculation of stress underlying a triangular load resting on an elastic half space as a very rough estimate, the maximum deviatoric stress occurs at 150-kin depth and equals about 200 MPa.

Thurber and Toksoz's [1978] model assumes that Olympus Mons is supported only by the strength of the Martian lithosphere. Phillips et al. [1980] argue that the implied stresses are too large and that the Tharsis region must be at least partially supported by dynamic forces from below.

There is even an indication that stress estimates based on a

volume measure of excess mass from Olympus Mons may be too low. From Viking Orbiter 2 high-resolution gravity data,

Sjogren [1979] fails to detect any low-density compensating mass for Olympus Mons. The gravity anomaly over the feature is about 20% greater than expected from the observed volume using a Bouguer theory and suggests that even denser material lies beneath the volcano. Sjogren concludes, 'This essentially uncompensated 600-km feature produces kilobar stresses that demand a rigid, thick lithosphere, or some rather unique scenario about very young topography obtained on a presently seismically inert planet.'

Smith et al. [1979] note a high correlation between gravity and topography on the planet Venus. Their data are best explained by a regional compensation mechanism with D -- 5 x 1023 N m. The corresponding 40-kin plate thickness is so similar to that determined from terrestrial studies that we might be tempted to conclude that the two planets have comparable strength and rheological properties in their surface layers. The observation that the temperature on the surface of Venus is

/DEFORMED LAYER 400øC complicates the interpretation. Watts [1978] has proposed that the base of the elastic layer in the earth corresponds to approximately the 450øC isotherm. At this stage the limitations in the data allow only speculation; if the same model applies to Venus, there are at least two possible ex-

ELASTIC LAYER planations for the compensation. Temperature may be a very slowing increasing function of depth beneath the Venusian surface, or the chemistry of the rocks might allow greater strength at higher temperatures.

Fig. 14. Schematic diagram showing two styles of tectonic deformation. (a) Shallow compressional forces deform weak sedimentary layer overlying more competent elastic plate. (b) Vertical or tensional forces disrupt entire thickness of competent layer.

CONCLUSIONS

At this point we must in some way answer the question, What magnitude stress can the lithosphere support on geologic time scales? Owing to the fundamental differences in oceanic and continental tectonic settings the oceanic loads tend to provide a good estimate of maximum stress level,


while the continents better indicate the duration of stress support. For example, from seismic refraction, surface gravity, and bathymetry studies, the spatial extent and magnitude of lithospheric strain caused by the Hawaiian load is rather well documented. There exists nowhere on the continents such a

classic example of lithospheric flexure. The continual overprinting of continental orogenic, volcanic, and epeirogenic episodes, with attendant erosion, obscures the deformation re- corded during any one event. On the other hand, since no oceanic loads are older than about 200 m.y., it is not possible to rule out relaxation throughout the thickness of the lithosphere on a time scale of 100 m.y. The 30-kin elastic plate thickness supporting the midcontinent gravity high [Cohen and Meyer, 1966] rather convincingly argues against lithospheric relaxation on the continents on billion-year time scales. While it is dangerous to apply oceanic stress estimates to the continents or assume that continental time scales are appropriate to the oceans, it appears that a mechanical lithosphere, 30-km or so thick, sustains stresses to about 200 MPa over billion-year time scales. The model results are not unique, in part owing to the ambiguity of gravity data and in part to the uncertainty in rheology, but it is unlikely that refinement in observations could drastically change these estimates. In particular, there is no conflict between empirical flow laws extrapolated from laboratory data and observed geophysical deformation data.

Acknowledgments. I Wish to thank T. C. Hanks, R. L. Parker, A. McGarr, J. COckran, and an anonymous reviewer for suggesting several improvements on an earlier form of this manuscript.

REFERENCES

Allen, R. R., Depth of sources of gravity anomalies, Nature Phys. Sci., 236, 22-23, 1972.

Anderson, R. N., D. P. McKenzie, and J. G. Sclater, Gravity, bathymetry and convection in the earth, Earth Planet. Sci. Lett., 18, 391-407, 1973.

ArtyUshkov, E. V., The stresses in the lithosphere caused by crustal thickness inhomogenieties, J. Geophys. Res., 78, 7675-7708, 1973.

Artyushkov, E. V., Can the earth's crust be in a state of isostasy?, J. Geophys. Res., 79, 741-752, 1974.

Ashby, M. F., and R. A. Verrall, Micromechanisms of flow and fracture and their relevance to the rheology of the upper mantle, Phil. Trans. Roy. Soc. London, Ser. A, 288, 59-95, 1977.

Balmino, G., K. Lambeck, W. H. Kaula, A spherical harmonic analysis of the earth's topography, J. Geophys. Res., 78, 478-481, 1973.

Banks, R. J., R. L. Parker, and J.P. Huestis, Isostatic compensation on a continental scale: Local versus regional mechanisms, Geophys. J. Roy. Astron. Soc., 51,431-452, 1977.

Beaumont, C., The evolution of sedimentary basins on a vis•lastic lithosphere: TheOry and examples, Geophys. J. Roy. Astron. Soc., 55, 471-497, 1978.

Beaumont, C., On rheological zonation of the lithosphere during flexure, Tectonophysics, 59, 347-365, 1979.

Bodine, J. H., and A. B. Watts, on the lithospheric flexure seaward of the Bonin and Mariana trenches, Earth Planet. Sci. Lett., 43, 132- 148, 1979.

Brune, J. N., and C. R. Allen, A low-stress drop low magnitude earthquake with surface faulting: The Imperial California earthquake of March 4, 1966, Bull Seisrnol. Soc. Arner., 57, 501-514, 1967.

Carter, N. L., High temperature flow of rocks, Rev. Geophys. Space Phys., 13, 344-349, 1975.

Carter, N. L., Steady state flow of rocks, Rev. Geophys. Space Phys., 14, 301-360, 1976.

Carter, N. L., and H. G. Av6 Lallement, High temperature flow of dunite and peridotite, Geol. Soc. Amer. Bull., 81, 2181-2202, 1970.

Chase, C. G., Subduction, the geoid, and lower mantle convection, Nature, 282, 464-467, 1979.

Cochran, J. R., An analysis of isostasy in the world's oceans, 2, Mid- ocean ridge crests, J. Geophys. Res., 84, 4713-4729, 1979.

Cohen, T. J., and R. P. Meyer, The mid continent gravity high: Gross crustal structure, in The Earth Beneath the Continents, Geophys. Monogr. Ser., vol. 10, edited by J. S. Steinhardt and T. J. Smith, pp. 141-165, AGU, Washington, D.C., 1966.

Dorman, L. M., and B. T. R. Lewis, Experimental isostasy, 1, Theory of determination of the earth's isostatic response to a concentrated load, J. Geophys. Res., 75, 3357-3365, 1970.

Gaposchkin, E. M., and K. Lambeck, Earth's gravity field to sixteenth degree and station coordinates from satellite and terrestrial data, J. Geophys. Res., 76, 4844-4883, 1971.

Goodacre, A. K., A comment on depths of sources of anomalies in the earth's gravity field, Geophys. J. Roy. Astron. Soc., 55, 745-746, 1978.

Griggs, D. T., F. J. Turner, and H. C. Heard, Deformation of rocks at 500 ø to 800øC, Geol. Soc. Amer. Mem. 79, chap. 4, 1960.

Guier, W. H., and R. R. Newton, The earth's gravity field as deduced from the Doppler tracking of five satellites, J. Geophys. Res., 70, 4613-4626, 1965.

Gunta: R., A quantitative evaluation of the influence of the lithosphere on the anomalies of gravity, J. Franklin Inst., 236, 373-396, 1943.

Hanks, T. C., The Kuril trench'-Hokkaido rise system: Large shallow earthquakes and simple models of deformation, Geophys. J. Roy. Astron. Soc., 23, 173-189, 1971.

Hanks, T. C., Earthquake stress drops, ambient tectonic stresses, and stresses that drive plate motions, Pure Appl. Geophys., 115, 330-331, 1977.

Hanks, T. C., Deviatoric stresses and earthquake occurrence at the outer rise, J. Geophys. Res., 84, 2343-2347, 1979.

Haxby, W. F., D. L. Turcotte, and J. M. Bird, Thermal and mechanical evolution of the Michigan Basin, Tectonophysics, 36, 57-75, 1976.

Higbie, J. W., and F. D. Stacey, Interpretation of global gravity anomalies, Nature Phys. Sci., 234, 130, 1971.

James, D. E., T. J. Smith, and J. S. Steinhart, Crustal structure of the m•iddle Atlantic states, J. Geophys. Res., 73, 1983-2008, 1968.

Jeffreys, H., The Earth, Cambridge University Press, New York, 1924. Jeffreys, H., The Earth, Cambridge University Press, New York, 1943. Jeffreys, H., The Earth, Cambridge University Press, New York, 1976. Kaula, W. M., Earth's gravity field, relation to global tectonics, Sci-

ence, 169, 982-985, 1969. Kaula, W. M., Global gravity and tectonics, in The Nature of the Solid

Earth, edited by E. C. Robertson, pp. 385-405, McGraw-Hill, New York, 1972.

Kaula, W. M., The moment of inertia of Mars, Geophys. Res. Lett., 6, 194-196, 1979.

Khan, M. A., Depth of sources of gravity anomalies, Geophys. J. Roy. Astron. Soc., 48, 197-209, 1977.

Kirby, S. H., State of stress in the lithosphere: Inferences from the flow laws of olivine, Pure Appl. Geophys., 115, 245-258, 1977.

Kirby, S. H., Tectonic stresses in the lithosphere: Constraints provided by the experimental deformation of rocks, J. Geophys. Res., 85, this issue, 1980.

Lambeck, K., Gravity anomalies over ocean ridges, Geophys. J. Roy. Astron. Soc., 30, 37-53, 1972.

Lerch, F. J., C. A. Wagner, J. A. Richardson, and J. E. Brownd, God- dard earth models (5 and 6), Doc. X-921-74-145, 100 pp., Goddard Space Flight Center, Greenbelt, Md., 1974.

Lewis, B. T. R., and L. M. Dotman, Experimental isostasy, 2, An isostatic model for the U.S.A. derived from gravity and topographic data, J. Geophys. Res., 75, 3367-3386, 1970.

Liu, H. P., and D. Kosloff, Elastic-plastic bending of the lithosphere incorporating rock deformation data, with application to the structure of the Hawaiian Archipelago, Tectonophysics, 50, 249-274, 1978.

McAdoo, D.C., J. G. Caldwell, and D. L. Turcotte, On the elastic- perfectly plastic bending of the lithosphere under generalized loading with application to the Kuril trench, Geophys. J. Roy. Astron. Soc., 54, 11-26, 1978.

McKenzie, D. P., Some remarks on heat flow and gravity anomalies, J. Geophys. Res., 72, 6261-6273, 1967.

McKenzie, D. P., Speculations on the causes and consequences of plate motions, Geophys. J. Roy. Astron. Soc., 18, 1-32, 1969.

McKenzie, D. P., Surface deformation, gravity anomalies and convection, Geophys. J. Roy. Astron. Soc., 48, 211-238, 1977.

McKenzie, D. P., J. M. Roberts, and N. O. Weiss, Toward the numer-


ical simulation of convection in the earth's mantle, J. Fluid Mech., 62, 465-538, 1974.

McNutt, M. K., Compensation of oceanic topography: An application of the response function technique to the Surveyor area, J. Geophys. Res., 84, 7589-7598, 1979.

McNutt, M. K., and H. W. Menard, Lithospheric flexure and uplifted atolls, J. Geophys. Res., 82, 1206-1212, 1978.

McNutt, M. K., and R. L. Parker, Isostasy in Australia and the evolution of the compensation mechanism, Science, 199, 773-775, 1978.

Melosh, H. J., Dynamic support of the outer rise, Geophys. Res. Lett., 5, 321-324, 1978.

Menard, H. W., Depth anomalies and the bobbing motion of drifting islands, J. Geophys. Res., 78, 5128-5137, 1973.

Mercier, J. C., and N. L. Carter, Pyroxene geotherms, J. Geophys. Res., 80, 3349-3362, 1975.

Murrell, S. A. F., Rheology of the lithosphere-experimental in- dications, Tectonophysics, 36, 5-24, 1976.

Nadai, A., Theory of Flow and Fracture of Solids, vol. 2, pp. 272-297, McGraw-Hill, New York, 1963.

Nunn, J. A., and N.H. Sleep, Gravity anomalies and flexure of the Michigan Basin: A three-dimensional study, Eos Trans. ,4 G U, 60, 936, 1979.

Phillips, R. J., and R. S. Saunders, The isostatic state of Martian topography, J. Geophys. Res., 80, 2893-2898, 1975.

Phillips, R. J., W. B. Banerdt, N. L. Sleep, and R. S. Saunders, Mar- tian stress distribution: Arguments about finite support of Tharsis, J. Geophys. Res., in press, 1980.

Reasenberg, R. D., The moment-of-inertia and isostasy of Mars, J. Geophys. Res., 82, 369-375, 1977.

Sclater, J. G., and P. A. F. Christie, Continental stretching: An explanation of the post mid-Cretaceous subsidence in the central North Sea basin, J. Geophys. Res., 85, 3659-3669, 1980.

Sclater, J. G., R. N. Anderson, and M. L. Bell, The elevation of ridges and the evolution of the central eastern Pacific, J. Geophys. Res., 76, 7888-7915, 1971.

Sclater, J. G., L. A. Lawver, and B. Parsons, Comparisons of long- wavelength residual elevation and free air gravity anomalies in the North Atlantic and possible implications for the thickness of the lithospheric plate, J. Geophys. Res., 81, 1031-1052, 1975.

Sjogren, W. L., Mars gravity: High resolution results from Viking Oc- tober 2, Science, 203, 1006-1009, 1979.

Smith, J. C., R. J. Phillips, and W. L. Sjogren, An analysis of Venus gravity data (abstract), Fall Meeting Program, p. 48, AGU, Wash- ington, D.C., 1979.

Solomon, S.C., The relationship between crustal tectonics and internal evolution in the Moon and Mercury, Phys. Earth Planet. Interi- ors, 51, 135-145, 1977.

Solomon, S.C., On volcanism and thermal tectonics on one-plate planets, Geophys. Res. Lett., 5, 461-464, 1978.

Solomon, S.C., and J. W. Head, Vertical movement in mare basins: Relation to mare emplacement, basin tectonics, and lunar thermal history, J. Geophys. Res., 84, 1667-1682, 1979.

Stephenson, R., and C. Beaumont, Small scale convection in the upper mantle and the isostatic response of the Canadian shield, re- port, Advan. Sci. Inst., Newcastle-Upon-Tyne, England, April 1979.

Suyenaga, W., Earth deformation on response to surface landing: Ap- plication to the formation of the Hawaiian Islands, Ph.D. dis- sertation, 147 pp., Univ. of Hawaii, Honolulu, 1977.

Thurber, C. H., and S.C. Solomon, An assessment of crustal thickness variations on the lunar near side: Models, uncertainties, and implications for crustal differentiation, Proc. Lunar Planet. Sci. Conf. 9th, 1169-1170, 1978.

Thurber, C. H., and M. N. Toksoz, Martian lithospheric thickness from elastic flexure theory, Geophys. Res. Lett., 5, 977-980, 1978.

Turcotte, D. L., D.C. McAdoo, and J. C. Caldwell, An elastic-perfectly plastic analysis of the bending lithosphere, Tectonophysics, 47, 193, 1978.

Walcott, R. I., Flexural rigidity, thickness, and viscosity of the lithosphere, J. Geophys. Res., 75, 3941-3954, 1970a.

Walcott, R. I., Flexure of the lithosphere at Hawaii, Tectonophysics, 9, 435-446, 1970b.

Walcott, R. I., Lithospheric flexure, analysis of gravity anomalies, and the propagation of seamount chains, in The Geophysics of the Pa- cific Ocean Basin and Its Margin, Geophys. Monogr. Ser., vol. 19, edited by G. H. Sutton, M. H. Manghnani, and R. Moberly, pp. 431-438, AGU, Washington, D.C., 1976.

Watts, A. B., Gravity and bathymetry in the central Pacific Ocean, J. Geophys. Res., 81, 1533-1553, 1976.

Watts, A. B., An analysis of isostasy in the world's oceans, 1, Hawai- ian-Emperor Seamount Chain, J. Geophys. Res., 83, 5989-6004, 1978.

Watts, A. B., and J. R. Cochran, Gravity anomalies and flexure of the lithosphere along the Hawaiian-Emperor Seamount Chain, Geophys. J. Roy. ,4stron. Soc., 38, 119-141, 1974.

Watts, A. B., and M. Talwani, Gravity anomalies seaward of deep-sea trenches and their tectonic implications, Geophys. J. Roy. ,4stron. Soc., 36, 57-90, 1974.

Watts, A. B., J. H. Bodine, and M. S. Steckler, Observations of flexure and the state of stress in the oceanic lithosphere, J. Geophys. Res., 85, this issue, 1980.

Weertman, J., The creep strength of the earth's mantle, Rev. Geophys. Space Phys., 13, 344-349, 1970.

Weertman, J., and J. R. Weertman, High temperature creep of rock and mantle viscosity, ,4nnu. Rev. Earth Sci., 3, 239-316, 1975.

Weissel, J. K., and D. E. Hayes, The Australian-Antarctic dis- cordance: New results and implications, J. Geophys. Res., 79, 2579- 2587, 1974.

Zienkiewicz, O. C., The Finite Element Method in Engineering Sci- ences, McGraw-Hill, New York, 1971.

(Received November 2, 1979; revised February 15, 1980; accepted March 6, 1980.)

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