Radial profile of mantle viscosity: Results from the joint inversion of convection and postglacial

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 102, NO. B2, PAGES 2751-2769, FEBRUARY 10, 1997 Radial profile of mantle viscosity: Results from the joint inversion of convection and postglacial rebound observables Jerry X. Mitrovica Department of Physics, University of Toronto, Toronto, Ontario, Canada Alessandro M. Forte D6partement de Sismologie, Institut de Physique du Globe de Paris, Paris Abstract. We present new inferences of the radial profile of mantle viscosity that simultaneously fit long-wavelength free-air gravity harmonics associated with mantle convection and a large set of decay times estimated from the postglacial uplift of sites within previously glaciated regions (Hudson Bay, Arctic Canada, and Fennoscandia). The relative sea level variation at these latter sites is constrained by age-height pairs obtained by geological survey, rather than the subjective trends which are commonly used in glacial isostatic adjustment (GIA) studies. Our viscosity inferences are generated using two approaches. First, we adopt a relative viscosity profile which is known to provide a good fit to the free-air gravity harmonics and determine an absolute scaling which yields a best fit to the GIA decay time constraints. Second, we perform an iterative, nonlinear, joint inversion of the two data sets. In both cases our inferred profiles are characterized by a significant increase of viscosity (-02 orders of magnitude), with depth, to values of -01022 Pa s in the bottom half of the lower mantle. The new viscosity profiles are shown to satisfy constraints based on the postglacial uplift of both Fennoscandia (the classic Haskell [1935] number) and Hudson Bay which have commonly been invoked to argue for an isoviscous mantle. Furthermore, the models are used to predict a set of long-wavelength signatures of the GIA process. These include predictions of GIA-induced variations in (1) the length-of-day over the late Holocene period; (2) the Earth's precession constant and obliquity over the last 2.6 Myr; and (3) the present-day zonal harmonics of the geopotential, j (l _< 7). The predictions (1) and (3) bound the late Holocene (and ongoing) mass flux between the large polar ice sheets (Greenland and Antarctic) and the global oceans to small values (<_ 0.4 mm/yr equivalent eustatic sea level rise). Introduction The fluid dynamics of the Earth's mantle is strongly influenced by the value and radial variation of viscosity within that region. For example, recent threedimensional numerical simulations of mantle circulation indicate that a significant increase in viscosity, with depth, results in a convective planform dominated rived from surface geophysical observables (e.g., longwavelength gravity anomalies and plate motions) associated with the convection circulation [e.g., l-lager, 1984; Ricard et el., 1984; Richards and l-lager, 1984; Forte and Peltlet, 1987, 1991; Ricard et el., 1989; Ricard and Vigny, 1989; l-lager and Clayton, 1989; Forte et el., 1991, 1993, 1994; King and Masters, 1992; Corrieu et el., 1994; King, 1995]. Historically, the first in situ inferences of viscosity were obtained from analyses of data associated with by long-wavelength, linear downwellings, akin to subduction zones [Zhang and Yuen, 1995; Bunge et. el, 1996]. Furthermore, the dynamic topography of plates glacial isostatic adjustment (GIA) [e.g., Haskell, 1935]. supported by mantle convective flow is a strong' func- These analyses have, to date, incorporated a globally tion of the depth variation of mantle strength [Gurnis, distributed set of sea level variations [e.g., McConnell, 1992]. Given this connection, it is not surprising that 1968; Cathles, 1975; Peltier and Andrews, 1976; Wu and inferences of mantle viscosity have commonly been de- Peltier, 1983; Nakada and Lainbeck, 1987, 1989; Wolf, 1987; Lambeck et al., 1990; Tushingham and Peltlet, Copyright 1997 by the American Geophysical Union. 1992; Mitrovica and Peltier, 1995; Han and Wahr, 1995; Paper number 96JB03175. Davis and Mitrovica, 1996] and anomalies in both the 0148-0227/97/96JB-03175509.00 Earth's gravity field [e.g., O'Connell, 1971; Cathles, 2751

2752 MITROVICA AND FORTE: JOINT INVERSIONS FOR MANTLE VISCOSITY 1975; Wu and Peltier, 1983; Rubincam, 1984; Mitrovica and Peltlet, 1989; Ivins et al., 1993; Han and Wahr, 1995] and rotational dynamics [O'Connelt, 1971; Nakiboglu et al., 1980; $abadini and Peltlet, 1981; Yuen et late Pleistocene ice load history and the limited class of viscosity models considered in most forward analyses of GIA (usually two-layer discretizations of the sublithospheric mantle). The former uncertainty has motivated al., 1982; Wu and Pettier, 1984]. Numerical predictions a search for data parameterizations which are insensiof GIA have, in turn, been applied to correct modern tide gauge records and estimate global sea level varitive to the surface load. Nakada and Lambeck [1989], for example, have advocated the use of differential sea level ations [e.g., Peltlet and Tushingham, 1989; Douglas, highstands between far-field (mainly Australian) sites, 1991; Trupin and Wahr, 1991; Davis and Mitrovica, 1996], to predict present-day three-dimensional crustal deformations [e.g., James and Lambert, 1993; Mitrovica et al., 1993, 1994a], and to calculate both Holocene paleotopography [e.g., Johnston, 1993; Peltlet, 1994] and perturbations to the orbital parameters of the Earth while Mitrovica and Peltlet [1995] and Mitrovica [1996] have considere decay times associated with postglacial uplift curves in previously glaciated regions. Given the computational requirements of the GIA problem, the limited search through viscosity model space is best dealt with by applying inversion procedures to the data (precession frequency, obliquity) during the Pleistocene [e.g., Parsons, 1972; Mitrovica and Peltlet, 1991a, 1995; [e.g., Dehant et at., 1990; Mitrovica et al., 1994b, 1996; Forte and Mitrovica, 1996; Mitrovica, 1996]. Pettier and Jiang, 1994; Ito et al., 1995; Mitrovica and The inversion of postglacial decay times have pro- Forte, 1995]. The numerical predictions required in vided some encouraging results toward a resolution of these applications and the many conclusions based upon the mantle viscosity problem. For example, Mitrovica them are extremely sensitive to the adopted radial pro- [1996] has found that the resolving power associated file of mantle viscosity. with the classic Haskell [1935, 1936] constraint on man- The first viscosity estimates based on long-wavelength tle viscosity ( 0102 Pa s) extends to about 1400 km geoid harmonics [e.g., Hager, 1984; Ricard et al., 1984; depth [see McKenzie, 1967; O'Connett, 1971], rather Richards and Hager, 1984] suggested a rather signifi- than 670 km depth, as has been presumed in many recant increase of viscosity, with depth, in the mantle. These estimates were, however, introduced during a period when the prevailing view from GIA was that the viscosity increased only moderately, if at all, fron the base of the lithosphere to the core-mantle boundcent analyses [e.g., Tushingham and Peltlet, 1992]. This misinterpretation was found to play a role in biasing many analyses toward near-isoviscous mantle models. Mitrovica [1996] has also argued that the insensitivity of most relative sea level (RSL) observables associated ary (CMB) (e.g., Cathles [1975], Pettier and Andrews with GIA, to variations in vis : )sity within the bottom [1976], Wu and Pettier [1983], continuing to Tushing- 1000 km of the mantle, ntay provide the degree of freeham and Peltlet [1992]). This disagreement motivated dom required to bring GIA inferences into accord with investigations of the influence of transient mantle vis- those based on convection observables. A recent joint cosity on GIA predictions [e.g., $abadini et at., 1985; inversion of long-wavelength convection and GIA ob- Pettier, 1985; Peltlet et al., 1986; Yuen et at., 1986]. servables [Forte and Mitrovica, 1996] has confirmed, in However, while intervening inferences from convection principle, that such a reconciliation is possible (see also data have confirmed the necessity for a large increase Hager [1991]). in mantle viscosity (see the above list of references), a Forte and Mitrovica [1996] reported results on a sigrowing number of GIA analyses have disputed the sug- multaneous nonlinear Occam inversion [Constablet at., gestion of a nearly isoviscous mantle [e.g., Nakada and 1987] of long-wavelength (up to degree 8), free-air grav- Lambeck, 1987; 1989; Lambeck et al., 1990; Mitrovica, ity data and decay times from two particularly accurate 1996; Forte and Mitrovica, 1996] and returned to an ear- RSL curves: Richmond Gulf in Hudson Bay and Angerlier view that the mantle viscosity increased markedly man River in Sweden. The small number of GIA data with depth [e.g., McConnell, 1968]. Several factors contribute to the discord amongst the GIA community. For example, inferences of mantle viscosity using GIA data have been based, with few exceptions, on spherically symmetric Earth models. It is clear that lateral heterogeneities in viscosity can influence the adjustment process [e.g., $abadini et al., 1986; Gasperini and $abadini, 1989; Manga and O'Connelt, 1995; Giunchi et al., 1996] and that these heterogeneities may introduce discrepancies in inused in the analysis renders it somewhat preliminary. Accordingly, in the present paper we extend the inversion to include 26 decay times scattered within Hudson Bay, Arctic Canada, and Fennoscandia. The new analysis confirms the essential features of our earlier inversion, namely, that the GIA and long-wavelength convection data sets can be reconciled by a radial viscosity profile characterized by significant ( 02 order of magnitude) increase of viscosity, with depth. We will demonstrate, in detail, how this increase can be accommodated while ferences that were based on different subsets of GIA still satisfying both the Haskell [1935] constraint on visdata. There are, however, many examples of disagree- cosity and the constraint provided by data from the ment arising from analyses based on data from the same Hudson Bay region alone (Mitrovica and Peltlet [1995] geographic region [e.g., McConnell, 1968; Fjeldskaar have shown that the average viscosity in the depth range and Cathtes, 1991]. Mitrovica and Peltlet [1995] and Mitrovica [1996] have argued that disparities in previous inferences may have arisen from uncertainties in the 400 to 1800 km depth is near i - 2 x 102 Pa s below this region). Finally, we consider the geophysical implications of our results by using the new models to predict

ß MITROVICA AND FORTE: JOINT INVERSIONS FOR MANTLE VISCOSITY 2753 present-day secular variations in the zonal harmonic of the Earth's geopotential (the so-called Jl harmonics and in particular 2), length-of-day variations through the late Holocene, and GIA-induced Pleistocene variations in the Earth's orbital parameters (precession, obliquity). These observables are chosen for two reasons. First, they represe.nt long-wavelength zonal degree-two signatures of the GIA process, and hence they are insensitive to the detailed geometry of the ice load history [e.g., Mitrovica and Peltlet, 1989; Mitrovica and Forte, 1995]. Second, the observables are also relatively insensitive to the lithospheric thickness [e.g., Mitrovica and Peltier, 1989; Mitrovica and Forte, 1995], which is a parameter left essentially unconstrained by our inversions. The inversions presented here do-not include two GIA data sets which might initially be considered useful. The first is the Fennoscandian relaxation spectrum derived by McConnell [1968]. The spectrum has been inverted by both Parsons [1972] and Mitrovica and Peltlet [1993a]; however, it is now clear that the uplift curves upon which it has been based are suspect [e.g., Wolf, 1996]. Inclusion of this data set which, in theory, is insensitive to the ice load history, awaits the derivation of a revised spectrum (K. Wieczerkowski et al., manuscript in preparation, 1996).. We also do not use observational constraints on the J2 harmonics [e.g., Eanes, 1995; Nerem and Klosko, 1996; Eanes and Bettadput, 1996] since these harmonics may have significant contribution, for example, from present-day melting events [$abadini et al., 1988; Trupin et al., 1992; Mitrovica and Peltlet, 1993b; Trupin, 1993; James and Ivins, 1995]. Instead, we will compare our a posteriori predictions of the J2 signal due to GIA to the observed value in order to derive constraints on the present-day mass flux between the ice sheets and the global oceans. This calculation will have the added benefit of drawing attention to, and resolving, recent inconsistent GIA predictions of these harmonics [Mitrovica and Peltlet, 1993b; Peltlet and Jiang, 1996]. Glacial Isostatic Adjustment' and Observables Theory Our forward predictions of GIA will be based on the response of a spherically symmetric, self-gravitating, Maxwell viscoelastic Earth model. In general, we adopt the usual viscoelastic Love number (i.e., normal mode) approach described by Peltier [1974], which is based on the application of the correspondence principle. Fang and Hager [1994] and Hanyk et al. [1995] have argued that this normal mode approach may have problems representing the continuous part of the response spectrum associated with our Earth model. Accordingly, we have verified our predictions (to an accuracy bet- ter than 5%) using a formalism in which the time domain inversion of the Laplace domain response is calculated, following the philosophy of Fang and Hager [1995], using a path integral in the complex s plane (rather than assuming a normal mode representation and limiting calculations to the negative real axis of this plane). The elastic structure of the Earth model is given by the seismic preliminary reference Earth model (PREM) [Dziewonski and Anderson, 1981]. The sea level equation governing gravitationally self-consistent ocean mass redistributions will be solved, up to degree and order 128, using the pseudospectral approach outlined by Mitrovica and Peltier [1991b]. It is well established that predictions of postglacial sea level variations near the center of previously glaciated regions are characterized by simple exponential forms [e.g., Mitrovic and Peltlet, 1995; Han and Wahr, 1995]. The same is true of accurate postglacial uplift curves from the same regions obtained by geological survey [e.g., Andrews, 1970]. As a model for this form, we adopt the single exponential function: RSL (t) = A [exp(t/r )- 1], where i denotes the ith geographic site, t is time measured into the past, and ri is the characteristic decay time of the exponential variation. A number of studies [e.g., Mitrovica and Peltlet, 1995; Forte and Mitrovica, 1996; Mitrovica, 1996] have shown that decay times determined from the postglacial uplift of some sites provide constraints on mantle viscosity that are remarkably insensitive to uncertainties in the ice load history. Once a postglacial time window for the analysis is decided upon (see below), a simple Monte Carlo approach is used to determine the best fitting pair (Ai,ri) for both predicted and observed RSL trends. In the case of the observed data we follow the approach of Mitrovica [1996] and augmenthe right-hand side of (1) with a site-dependent constant. This addition is important because many observed RSL curves are uncertain to a constant datum shift. In the present analysis we consider all sites in Hudson Bay, Arctic Canada, and Fennoscandia included in a recent global database of RSL curves [Tushingham and Peltier, 1992]. The locations of these sites are shown in Figures la and lb. As a first step, we determine an appropriate time window for the analysis. Mitrovica and Peltlet [1995] have argued that this window should be chosen to reflect the local postglacial regime in order that elastic effects do not contaminate the decay time estimates. To this end, they used only the last 6.5 kyr B.P. of data in their analysis of uplift curves from Hudson Bay. For the same reason, Mitrovica [1996] and Forte and Mitrovica [1996] adopted a time window of 9 kyr to analyze RSL variations at Angerman River and Oslo. (The longer time window reflects an earlier completion of the deglaciation event in Fennoscandia relative to the ice retreat from Hudson Bay.) If we are to extend the analysis to consider, at least as a first step, all the sites in Figures la and lb, then somewhat more care must be taken to select the appropriate time window. Previous analyses [Mitrovica and Peltier, 1995; Forte and Mitrovica, 1996; Mitrovica, 1996] have focused exclusively on sites near the center of either Laurentia or Fennoscandia, and therefore the predicted (and observed) RSL amplitudes were very large. This ( )

- 2754 MITROVICA AND FORTE: JOINT INVERSIONS FOR MANTLE VISCOSITY 2OO 22O 6O I 200 220 60 220 320 / A / I x L 0 20 40 7O 6O /D I / 70 a--... 60 the form (1) for the period subsequent to the kink. We have repeated these calculations for all of the sites in Figures l a and lb and for a set of global ice models and have concluded that a time window of the last 5.0 kyr B.P. for Fennoscandian sites and 6.5 kyr B.P. for sites in Hudson Bay and the Arctic will ensure robust decay time estimates that are unbiased by either elastic deformation or eustatic sea level variations. Although we have chosen to consider RSL sites that are included in the database of Tushingham and Peltier [1992], we will generally not use the "data" that appear in that compilation. The reason is that the compilation often replaces the actual data obtained by geological survey with RSL "trends" which are intended to reflect the scatter of the geological observations. To be specific, Tushingham and Peltlet [1992] have commonly elected to use the following procedure: (1) the geological data are plotted in the form of the usual RSL curves (age versus height of the samples); (2) an envelope is drawn that encompasses these constraints; (3) a set of quasidata are established at integer (Carbon-14) time intervals, which reflect the spread of the envelope in step 2 260 280 10 20 Figure 1. (a) and (b) The location of relative sea level (RSL) sites included in the database of Tushingham and Peltier [1992]. The squares and triangles denote sites considered in Figures 2 and 5, respectively (one site, Sam Ford Fiord, appears in both figures and thus has overlapping symbols). (c) and (d) The location of 26 sites culled from Figures la and lb according to guidelines described in the text. In the Tushingham and Peltier [1992] compilation sites A-O and a-k have the site number identifiers (101, 104, 106, 107, 108, 117, 125, 126, 127, 130, 133, 143, 144, 145, 152, 202, 209, 219, 221, 222, 224, 226, 227 228, 232, and 233) The reader is referred to Tushingham [1989] for detailed ref- erences associated with each site. 80 60-- Sam Ford Fiord (122) 40-- 20-- 0 3O 20- Skjaeafassen (204) - 60 Truelove Inlet (143) ß 45 30 15 30 0 I 20 _. Goteborg (227) would not be the case for all the sites in Figures la and lb, particularly those close to the edge of their respec- 10-- lo tive ice complexes at last glacial maximum. We must therefore consider, in choosing our time window, the potential contamination of these low-amplitude uplift o 0 : curves by global eustatic sea level fluctuations. lo 5 0 10 5 0 As an example, consider the numerical predictions of RSL variations in Figure 2, which includes two sites in the Arctic and two sites in Fennoscandia. The predic- Time (kyr B.P) Figure 2. RSL predictions at four sites near the edge tions are based on the ICE-3G global model for the of the maximum late Pleistocene ice cover over Canada final late Pleistocene deglaciation event [Tushingham (Sam Ford Fiord, Baitin Island and Truelove Inlet, Deand Peltier, 1991] and the specific viscoelastic model von Island) and Fennoscandia (Skjaeafassen, Norway described in the caption. Each of the curves exhibits and Goreborg, Sweden). The predictions are based on the ICE-3G model for the final late Pleistocene deglaciaa significant kink that is associated with the influence tion event and an Earth model characterized by a lithoof eustatic sea level variations produced by the ICEspheric thickness of 120 km and constant upper and 3G melting event. In the case of the Fennoscandian lower mantle viscosities of 1021 Pas and 2 x 1021 Pa s, sites (Skjaeafassen and Goreborg) the contamination respectively. Numbers at the top refer to the site idenbecomes evident at about 5 kyr B.P., while the same is tifier appearing in the compilation of Tushingham and true at approximately 6.5 kyr B.P. for the Arctic sites. Peltlet [1992]. See Figures la and lb (squares) for site The curves exhibit a simple single-exponential decay of locations.

MITROVICA AND FORTE: JOINT INVERSIONS FOR MANTLE VISCOSITY 2755 6o 40---- Tay Sound, Baffin Island (124) Time (kyr B.P) on the decay time (this bound will depend on the time window sampled by the geological observations). These Figure 3. The q-lrr observational errors on RSL age- cases are indicated by the arrows at the top of Figure height pairs (rectangles) obtained by geological survey 4a. at Tay Sound [Andrews and Drapier, 1967]. Crossbars represent q-lrr errors for the same site appearing in the The observational uncertainties in Figure 4a are concompilation of Tushingham and Peltier [1992](site 124). structed in two steps. First, an error in the decay The dashed line is the best fitting exponential form (1) time for a given site is computed by performing a least through the the four crossbar data points younger than squares solution starting at the best fitting form (1) 6.5 kyr (decay time r = 1.9 kyr, and amplitude A: 1.5 m). The dotted line is the best fitting form through the data obtained by survey (r = 19.2 kyr, A = 97.0 m). at these specific integer times; and (4) these quasi-data are then reported in the database. There are a number of problems associated with this procedure, particularly in regard to our present effort at establishing decay times. First, large temporal gaps exist in the geological data, and this requires exten- sive (and subjective) interpolation [see Mitrovica and Peltier, 1995]. Second, the decay times associated with the RSL trends may not be an accurate representation of the true observational constraints. As an example, consider the case of Tay Sound in Figure 3. The rect- The geological data appearing in Figure 3 do not represent upper and lower bounds on the RSL variation [see Andrews and Drapier, 1967], although the trend established by the quasi-data appears to have interpreted them as such. Nevertheless, geological constraints do commonly appear as upper and lower bounds. Such bounds, when they exist, are incorporated into our Monte Carlo procedure for determining the best fitting decay times. The full data set associated with the sites in Figures la and lb has been culled to remove RSL curves which significantly misfit the form (1) (Tay Sound in Figure 3 is an example), have less than two data points within the adopted time window, or have an observational error so large that they have no weight in the inverse procedures described below. The 26 sites that remain are shown in Figures lc and ld. Decay times determined for these sites are plotted in Figure 4a. A number of the RSL curves are characterized by nearly straight lines, and in these cases we can only determine a lower bound through the observations. Next, results from a suite of forward predictions, in which the ice history and lithospheric thickness are varied, were used to augment these errors to reflect uncertainties in both these inputs. Sites near the center of previously glaciated regions (for example, sites A-E in Hudson Bay (see Figure lc) and sites j-k (see Figure l d) in central Sweden) tend to have smaller observational uncertainties. There are two reasons for this. First, the RSL signal at these sites is comparatively large (as much as several hundred meters over the adopted time window). Second, the predicted decay times become more sensitive to variations in the lithospheric thickness and ice model as one moves from central sites to those closer to the edge of the ice complex at last glacial maximum [e.g., Mitrovica, 1996]. angles representhe geological constraints (taken frorn Mitrovica and Peltier [1995] inverted decay times Andrews and Drapier [1967], as referenced by Tushing- from six sites in Hudson Bay (Richmond Gulf, site A in ham [1989]). The five error bars (i.e., "quasi-data") Figure lc; Cape Henrietta Maria; Ottawa Island, site plotted at integer C-14 time (converted to sidereal time C; Southampton Island, site D; Ungava Peninsula, site in Figure 3) reflect the RSL trend chosen by Tushing- E; and James Bay). They pointed out that decay times ham and Peltier [1992]. The dashed line is the best from these sites could not all be fit by a single, spherifitting function of the form (1) through the four quasi- cally symmetric viscosity model (see also ttan and Wahr data points lying within our 6.5 kyr time window (all [1995]). This inconsistency is also apparent in Figure but the oldest quasi-data point are used in the Monte 4a, where the decay times for sites A, C, D, and E Carlo fit). The decay time for this form is 1.94-0.7 kyr vary by as much as a factor of 4 across a rather moder- (and the amplitude A - 1.5 m). The actual RSL data ate geographic range (see Figure lc). Mitrovica [1996] are, however, better fit (but not particularly well fit; extended the analysis of Mitrovica and Peltier [1995] see below) by the exponential form (1) with an order of to include two sites in Fennoscandia (Angerman River, magnitude longer decay time (19.2 kyr) and A = 97.0 m Sweden, site k in Figure ld; and Oslo, Norway, site f) (dotted line, Figure 3). In order to avoid these difficul- considered by Haskell [1935]. ties, we adopt, in the analyses below, the data obtained In the calculations described below we generally adopt by survey. a simplified ice model (hereinafter referred to as the

_ 2756 MITROVICA AND FORTE: JOINT INVERSIONS FOR MANTLE VISCOSITY 1.0 --.5-1.0-- T _.0 I I I I I I I I 1.0 c.0 I I I I I I! I, A B C D E F G ti I J K L M N O a b c d e f g h i j k Figure 4. Observational constraints on the decay times at 26 sites shown in Figures l c and l d (vertical solid lines). The constraints with an up-directed arrow represent sites at which only a lower bound on the decay time could be determined. For all other sites the verticalines represent q-l r error bars. The horizontal solid lines on each frame represent decay time predictions based on the DISK ice load, a gravitationally self-consistent ocean mass redistribution, and viscosity models (a) TP0 (dashed line, Figure 6a), (b) FDW-GIA (Figure 6b, left ordinate axis), and (c) MF2 (solid line, Figure 6a) (see text for model descriptions). The horizontal dotted lines are predictions based on the same Earth models, with the exception that an elastic lithosphere of thickness 120, 80, and 80 km, respectively, for each frame, has been included in the calculation. Site "DISK" model) that consists of a single disk load (elliptical in horizontal cross section and parabolic in vertical cross section) over each of Fennoscandia, Laurentia, and Arctic Canada. As an example of a set of forward calculations, in Figure 4a we show decay time predictions generated using the DISK ice load history, a gravitationally self-consistent ocean load [see Mitrovica and Peltlet, 1991b], and a viscosity model characterized by constant upper and lower mantle values of 102 and 2 x 102 Pa s, respectively. The solid and dashed horizontal lines refer to the cases of no litho- sphere (hereinafter referred to as model TP0), and an elastic lithosphere of thickness 120 km (model TP120), respectively. The latter model has been advocated by Tushingham and Peltlet [1992] on the basis of a global database of RSL histories. With the exception of the RSL site at Alert (site O), located on the northern tip of Ellesmere Island (and at the edge of the local ice complex), the predictions are relatively insensitive to the thickness and even existence of an elastic lithosphere. Models TP0 and TP120 perform reasonably well in fitting many decay times. However, these models yield decay times that are too long to fit both the totality of the Hudson Bay decay times (notice the large misfit to the Ottawa Island (site C) and Ungava Peninsula (site E) decay times) and the accurate decay time constraints from central Fennoscandia (Stockholm(site j) and Angerman River (site k)). One of the required inputs into the inversion procedure described below is the set of Frechet kernels (FK)

MITROVICA AND FORTE' JOINT INVERSIONS FOR MANTLE VISCOSITY 2757 for forward predictions of the decay time estimates. These kernels provide a measure of the sensitivity of the predictions to arbitrary depth-dependent variations in radial viscosity profile u(r). We can write: - (2) MB/a where r is the radius (nondimensionalized using the mean surface radius a). Figure 5 shows decay time kernels for four representative sites in Figures la and lb (denoted by the triangles), computed using the DISK ice load history and the TP0 viscosity model. Richmond Gulf and Angerman River are located near the center of the now-vanished Laurentide and Fennoscandian ice complexes, while Sam Ford Fiord and Lista are closer to the maximum perimeter of these respective regions. The area under the kernels in the lower and upper mantle have the ratios-- 2:1, 4:1, 1:1.2, and 1:2.5, as one moves down Figure 5. Since the Laurentide ice sheet was much larger than the Fennoscandian ice complex, the decay times from Canadian sites are more sensitive to lower mantle structure relative to upper mantle structure than those from Fennoscandia. Furthermore, in any given region the sensitivity of the datum mi- 4 Depth (km) 2500 1500 500 Richmond G If, Quebec I o/ I I s I ' I I Sam Ford Fiord, Baffin Is. 4 I,I o ' I ' I 1 - _ 4 Lista' Nørway,, r i o ' ]' 'l ] l, I Angerman River, Sweden 1 I [-t 4 [.MB 670 km 3400 4400 5400 6400 Radius (km) Figure 5. Frechet kernels (equation(2)) for the numerical prediction of the decay time associated with postglacial RSL variations at various sites (denoted by triangles in Figure 1) in Canada and Fennoscandia. The calculations adopt the TP0 viscosity profile given in Figure 6a (dashed line). grates to greater depths as one moves from "central" to "edge" sites. Note that the kernels all tend to zero in the deepest regions of the mantle and that the sign of the kernels is everywhere positive. The latter indicates that an increase in viscosity anywhere in the mantle will produce an increase in the decay time. Dynamically Supported Long- Wavelength Gravity Anomalies Numerous studies [e.g., Hager and Clayton, 1989; Forte and Peltlet, 1987, 1991; Ricard and Vigny, 1989] have previously demonstrated that the density perturbations derived from global models of three-dimensional (3D) mantle seismic heterogeneity may be used in mantle flow predictions to yield good fits to convectionrelated surface data such as the nonhydrostatic geoid and tectonic plate motions. Among these data, the nonhydrostatic geoid is especially sensitive to the rheological properties of the mantle [e.g., Ricard et al., 1984; Richards and Hager, 1984] and thus has provided the strongest constraints on the depth variation of viscosity governing the mantle convection process. The prediction of the nonhydrostatic geoid is based on the theory of buoyancy-induced flow in a spherical, self- gravitating mantle that is assumed to be either incompressible [e.g., Richards and Hager, 1984] or compressible [e.g., Forte and Peltier, 1991]. In the present study we employ the compressible flow theory throughout. We may summarize the flow theory by noting that the relationship between density perturbations in the mantle that drive convection and the spherical harmonic coefficients (degree œ and order m) of the free-air gravity anomalies G may ultimately be expressed in terms of a kernel function G using the equation G? = go 3(/?- 1) Gt[v(r)/Vo; r] 5p?(r) dr, (3) (2 + ) n where = 5.515 Mg/m a is the Earth's mean density, go = 9.82 m/s 2 is the mean surface gravitational acceleration, and 5p (r) are the harmonic coefficients of the density perturbations in the mantle. The/?-independent factor appearing before the integral in (3) arises from the relationship between the free-air gravity anomalies and the nonhydrostatic geoid [Forte et al., 1994] and from expressing the gravitational constant G in terms of Earth's mean surface gravity and mean density. In (3) we emphasize that the gravity kernels Gt are dependent not on the absolute viscosity but, rather, on the depth variation of relative viscosit y(r)/yo, where o is a reference scaling value. This contrasts with the strong dependence of the GIA data on the absolute value of mantle viscosity. To apply the mantle flow theory, we begin with perturbations of seismic velocity 5v/v obtained from tomographic models and convert these to density perturbations 5p using a velocity-to-density scaling coefficient 5 In p/5 In v. In all calculations presented in this article we use the shear wave velocity model S.F1.K/WM13 de-

2758 MITROVICA AND FORTE: JOINT INVERSIONS FOR MANTLE VISCOSITY rived by Forte et al. [1994]. Furthermore, estimates of yielded a variance increase of a factor of 16.9. Indeed, 5 lnp/5 In v$ based on theoretical [e.g., Karato, 1993], we found no case in which the TP0 model produced a laboratory [e.g., Chopelas, 1992], or geodynamicon- variance reduction, and we conclude that this viscosity siderations (see Forte et al. [1993, 1994] for a detailed model is entirely inadequate from the point of view of review) indicate that the conversion coefficient appro- the long-wavelength, free-air gravity data. priate for the mantle lies in the approximate range On the basis of a large suite of forward calculations, Forte et al. [1993; Figure 10c] have derived a relative +0.1 < 5 In p/5 In vs < + 0.4. (4) viscosity profile u(r)/uo, which performs well in fitting In our analyses we use the harmonic coefficients of the the long-wavelength, free-air gravity data. A 13-layer free-air gravity, rather than the coefficients of the corre- version of this model is shown in Figure 6b, where the sponding geoid, because the former have the advantage right-hand ordinate axis reflects the constraint on the of providing an evenly balanced representation of all relative, rather than absolute, viscosity (this axis adopts wavelengths in the large-scale geopotential (this is in the viscosity of the first mantle layer below 670 km contrast to the nonhydrostatic geoid, which is strongly depth to be the reference value Uo). This model (heredominated by its degree 2 and 3 harmonics) [Forte et inafter referred to as FDW) yields a 76% variance real., 1994]. As an example of a forward calculation duction to the free-air gravity harmonics when Karato's and following Forte and Mitrovica [1996], we have pre- [1993] velocity-to-density scaling profile is adopted. dicted free-air gravity harmonics using the TP0 model The Frechet kernels for the free-air gravity harmonics for a number of different velocity-to-density conversions are defined similarly to those for the decay times in (2) 5 In p/5 In rs. The variance increased by a factor of 7.6 when a constant scaling of 0.2 was adopted. The conver- FK [,(r)/,o; r]slog,(r) dr. (5) sion suggested by Karato [1993](in which 51np/5 In v$ MB/a = 0.2 at the surface, changing smoothly to 0.4 at 670 km depth, and returning smoothly to 0.2 at the CMB) In Figure 7 we show the gravity Frechet kernels for several harmonic coefficients computed using the same model (TP0) used to generate Figure 5. The kernels for Depth (km) the longest-wavelength gravity anomalies (e.g., œ- 2) 2500 1500 5OO are most sensitive to variations of viscosity in the bot- I A tom portion of the lower mantle. This region is where 22 the GIA decay time data provide little constraint on mantle rheology. At shorter wavelengths (e.g., œ = 5) the gravity data become equally sensitive to viscosity variations in the depth range 200-1000 km, and at 21 the shortest wavelengths considered here (œ = 8) they are most sensitive to viscosity variations in the middle portion of the upper mantle. It is thus evident 2O that the longest-wavelength gravity data provide complementary sensitivity to the GIA decay data, while at -1 intermediate and short wavelengths their sensitivity will overlap that of the decay times. Although the FDW viscosity provides a good fit to the convection observables, we have no guarantee that it is also compatible with the independent set of decay time data shown in Figure 4a. In the search for models of mantle viscosity that satisfy both the GIA 2o $ 3400 I ' I 4400 5400 Radius (km) 6400-1 data and the long-wavelength, free-air gravity harmonics, there are two approaches that are possible: (1) beginning with any geoid-inferred relative viscosity profile,(r)/,o, search for a scaling viscosity 'o such that the absolute viscosity,(r) fits the GIA data; or (2) simultaneously invert the two data sets to objectively determine an absolute viscosity profile that fits both. Each approach will be considered below. Figure 6. Viscosity profiles discussed in the text. (a) The starting (dashed line, model TP0), first iteration (dotted line, model MF1), and final (solid line, model MF2) profiles associated with the joint (Occam) inversion of long wavelength, free-air gravity data and postglacial decay times. (b) The viscosity profile FDW inferred by Forte et al. [1993]. The absolute viscosity is Inverse Analyses unconstrained the Forte et al. [1993] analysis (see the right-hand ordinate scale); the left-hand axis is the scaling (to model FDW-GIA) that provides a best fit to In approach 1 we seek a scaling viscosity 'o such that the data set of postglacial decay times. Model FDW- the absolute viscosity,(r) then delivered by the FDW GIA has a value of 18.7 in the layer just above 670 km relative viscosity profile best fits the GIA decay time depth. constraints. For reasons described below, our "fits" will

-- MITROVICA AND FORTE: JOINT INVERSIONS FOR MANTLE VISCOSITY 2759 o =2 (, l = (b) l =8 (e) 500 ' 1 ooo 1500 R 2000 I 2500 3000 I I i I I 1-60 o 60 I I I I I I i I I I I I -60 0 60-60 0 60 AMPLITUDE [regals] Figure 7. Frechet kernels for the numerical prediction of the free-air gravity harmonics for various spherical harmonic degrees œ- (a) 2, (b) 5, and (c) 8. The calculations adopt the TP0 viscosity profile given in Figure 6a (dashed line) and the velocity-density scaling 5 In p/5 In vs - +0.2. The kernels are'identified by the labels /E and I/, which correspond to the real and imaginary parts of FK, respectively, defined in (5). The dashed horizontaline identifies the location of the 670-km seismic discontinuity. be defined in terms of the logarithm of the decay times. ysis of Forte et al. [1991], the inversions are parame- Furthermore, we do not include lower bound constraints terized in terms of the logarithm of viscosity in a set on the decay times (e.g., site a in Figure 4) in determin- of 13 constant-viscosity layers (as in Figures 5, 6b, and ing these fits, but rather, we compare our model results 7) extending from the CMB to the solid surface. The against these bounds. Using a simple, one-parameter Occam inversion of the nonhydrostatic geoid data by inversion, followed by a suite of forward calculations, we have found that o - 1.6 x 102 Pa s applied to the FDW model minimizes the fit to the decay time data set. The inferred absolute viscosity axis is given by the left-hand ordinate scale in Figure 6b. The decay times predicted by the model (hereinafter referred to King and Masters [1992] was parameterized in terms of absolute viscosity and required numerous iterations. A representation in terms of the logarithm of the viscosity automatically ensures a positive-definite viscosity, and moreover, it renders the inverse problem quasilinear [Forte et al., 1991], thus ensuring relatively rapid as FDW-GIA) are shown in Figure 4b (solid horizontal convergence to a solution. Mitrovica and Peltlet [1995] lines). The FDW-GIA model yields an impressive fit to the decay time constraints. In contrast to the TP models (Figure 4a), the new model matches the decay times from central Fennoscandia and yields decay times that are roughly the average (in log space) of the highly variable Hudson Bay constraints. Furthermore, the fits to other RSL sites are comparable to those obtained using the TP models. The dotted lines in Figure 4b represent the decay times predicted by altering model FDW-GIA to have an have shown that inverting for the logarithm of the decay times also renders GIA inversions quasi-linear. As mentioned, the gravity and GIA data are scaled by their respective l a error estimates prior to inversion. To estimate for the gravity data, we assume that the uncertainty is not in the observations themselves but, rather, is entirely due to incomplete resolution of geodynamically important regions in the global 3-D seismic models. Indeed, as shown by Forte et al. [1994], we can achieve essentially perfect fits to the long-wavelength 80-km elastic lithosphere. As in Figure 4a, the thickness convection data with 3-D mantle heterogeneity models of a lithospheric plate has little effect on the predicted that still satisfy the global seismic data. This implies decay times, and therefore our inference of viscosity is that we can estimate the uncertainties in the free-air relatively insensitive to this aspect of the Earth model. Having now verified the existence of a viscosity profile that simultaneously fits both GIA decay data and longwavelength, free-air gravity data, we carry out formal gravity data (arising from poorly resolved 3-D seismic structure) by simply using the misfit between the data and the free-air gravity predictions previously calculated on the basis of a 3-D mantle model derived onl joint inversions of both data sets. Our nonlinear, itera- from seismic data. This misfit then provides an intive, viscosity inversions are performed using the "Oc- dependent a priori estimate of the data uncertainties cam" algorithm, described by Constablet al. [1987]. that can be used in the inversions. We therefore em- The Occam algorithm weights the individual data di ploy the misfit between the free-air gravity data (in the (i- 1,..., N) by their respective uncertainties ai. The degree range œ- I- 8) and the predictions obtained objective is then to find the smoothest possible model that yields a (statistically) acceptable fit to the data, on the basis of the previously published 3-D seismic model SH8/WM13 [Woodward et al., 1993] (and using corresponding to a X 2 misfit of N. Following the anal- the FDW viscosity model) to estimate a mean a of 0.86

2760 MITROVICA AND FORTE: JOINT INVERSIONS FOR MANTLE VISCOSITY mgal per spherical harmonic coefficient of the gravity data. We employ a constant er to weight the gravity data because, as noted above, the long-wavelength, freeair gravity data are characterized by a nearly fiat amplitude spectrum (unlike the geoid). A constant er thus implies that the shortest-wavelength data (at œ = 8) constrain the weighted average viscosity down to about 1400 km depth. Mitrovica [1996] has shown that this average (in log space) must lie in the range 20.90 q-0.10 to satisfy the Angerman River decay time. When both (and only) Angerman River and Goteborg are considered, then this inferred average drops to 20.73 q- 0.09. are given as much importance as the longest-wavelength Applying the averaging kernel to the model MF2, for exdata (œ = 2). We thereby ensure the upper mantle resolving power of the short-wavelength data (see Figure ample, yields a value of 20.75. The same weighting applied to the TP0 model yields 21.10, which is 4er higher 7) is retained in the inversions. than the observational constraint. Mitrovica and Peltlet In the inverse calculations the gravity anomaly data [1995] have argued that if the decay times from Hudson are interpreted in terms of a velocity-density conversion Bay are considered together, then they are best recon- coefficient 5 In p/5 In vs = 0.2. The use of other values for the velocity-density conversion appears to have a relatively small impact on the final inverted viscosity profiles. For example, when we employ 5 In p/5 In vs = 0.4, we find that the main difference is a slight reduction in the overall slope of the viscosity profile between 600 and 1000 km depth. Otherwise, the depth variation of the viscosity profile is very similar to that obtained for decay time data sets for both regions simultaneously, our inversion (to model MF2) acts to weaken the upper mantle relative to the TP value of 102 Pa s (see the 5 In p/5 In vs - 0.2. The main effect of assuming differ- following paragraph) and stiffen the top 1000 km of ent values for the velocity-density conversion is on the the lower mantle. The same trend is evident in model final X 2 misfit to the gravity data. Significantly, we have FDW-GIA, and it explains the success of that model found that two iterations are essential for convergence in fitting the decay times. Clearly, the decay times for to a model which simultaneously fits both the decay time and long-wavelength, free-air gravity harmonics. The starting (TP0) model, as well as the first and second iterate solutions (hereinaftereferred to as MF1 these regions do not require a nee rly constant, 102 Pa s viscosity in the top half of the mantle, as has been suggested by others [Peltier and Jiang, 1996, p. 3288]. The detailed structure within the upper mantle is and MF2) of the joint inversion, are shown in Figure forced primarily by constraints associated with the long- 6a. The decay times predicted using the MF2 model are shown in Figure 4c. As in the case of the FDW-GIA model, the model MF2 yields decay times that better fit (relative to the TP0 model) the decay times from central Fennoscandia and the totality of the constraints wavelength convection observables. As an example, the low-viscosity transition zone appears not only in model FDW-GIA and MF2, but also in other independent inferences based on convection observables [e.g., King and Masters, 1992; Corrieu et al., 1994]. The viscosity in from Hudson Bay. We have also verified in Figure 4c the bottom 1000 km of the lower mantle is also con- (dotted horizontal lines) that the MF2 inference is insensitive to any reasonable assumptions in regard to the strained, in large part, by the long-wavelength gravity data. The average viscosity within this region is thickness of an elastic lithosphere. 1022 Pa s in both the FDW-GIA and MF2 models. The first iteration model, MF1, fits the decay time The net effect of these various sensitivities is that our data equally as well as model MF2. The second iteration is required, however, to improve the model fit to inferences are characterized by a large increase in viscosity ( 2 orders of magnitude) from the upper mantle the long-wavelength gravity harmonics. Recall, that the TP0 model yielded a variance increase of a factor of 7.6 with respecto the free-air gravity data (for our adopted velocity-to-density conversion of 0.2). Models MF1 and MF2 improve this to variance reductions of 44% and 70%, respectively. When the velocity-to-density conversion described by Karato [1993] is adopted, then the model MF2 provides a variance reduction of 76%. This fit is the same as that obtained using the model FDW (as discussed previously). The question arises as to why the FDW-GIA and MF2 models yield improved fits to the decay time data in central Fennoscandia (in particular, Goteborg (site j) and Angerman River (site k)) and Hudson Bay relative to the TP0 (or TP120) models. The answer is best understood in terms of the resolving power of these decay data. The Angerman River kernel in Figure 5 indicates that the decay times from central Fennoscandia ciled with a model having a weighted average, defined by the Richmond Gulf kernel in Figure 5, near 21.0. This average, which has nonnegligible weighting down to depths near 2000 km, is 21.14 for the MF2 model and 21.20 for the TP0 model. Therefore, to satisfy down to the CMB (Figure 6). The viscosity inferences shown in Figure 6 are quite similar to those obtained by Forte and Mitrovica [1996], who reported on a joint inversion using only two decay times (Richmond Gulf (site A) and Angerman River (site k)). The main difference in the new results is that the absolute viscosity scale has been shifted downward by a factor of 1.3 (or -0.14 in log space). The reason for this is apparent from Figure 4. The Richmond Gulf decay time is longer than the "mean" decay time which characterizes the Hudson Bay sites. Furthermore, the Angerman River decay time is longer than that of Goteborg (site j, the only other site in central Fennoscan- dia). Thus reconciling the totality of the decay time estimates, while still maintaining a good fit to the longwavelength gravity observations, can be achieved (in an approximate sense) by reducing the reference viscosity o preferred by Forte and Mitrovica [1996].

MITROVICA AND FORTE' JOINT INVERSIONS FOR MANTLE VISCOSITY 2761 Geophysical Implications ever, the FDW-GIA model of Figure 6b, which exhibits no such feature, fits both the long-wavelength, free-air Our inferences of a large increase of mantle viscosity gravity harmonics and decay time data, just as well as with depth are consistent with an established and growthe model obtained by joint inversion of these data sets. ing body of work associated with the mantle convec- In any event, our results indicate that a steep viscostive circulation [e.g., Hailer, 1984; Ricard et al., 1984; ity increase at...1000 km depth cannot be ruled out by these data sets. Richards and ttailer, 1984; Gumis and Davies, 1980]. For example, it is generally accepted that the relative In the remainder of this section we explore the imstationarity of the hot spot reference frame requires plications of our viscosity models for a set of GIA prehigh viscosity in the deep mantle [e.g., Richards, 1991; dictions. Since our inferences do not place strong con- Steinberiler, 1990]. A stiff deep mantle region is also instraints on the thickness of a high-viscosity lithosphere, dicated by the observed long-term rates of polar wander, we avoid observables that are sensitive to this paramewhich suggest a slow advection of deep mantle density ter (e.g., RSL variations in the periphery and far field of the late Pleistocene ice sheets, present-day threeheterogeneities [e.g., Sabadini and Yuen, 1989; Spada et al., 1992; Steinberiler, 1990]. Within the upper mantle dimensional crustal deformations, etc.) In particular, region of weak viscosity has not only been inferred from we consider various long-wavelength anomalies associated with GIA. We nevertheless adopt a lithospheric long-wavelength geoid constraints [e.g., Kinil and Masthickness of 80 kin. ters, 1992; Forte et al., 1993; Corrieu et al., 1994], but is also suggested by the near-surface deflection pattern of Pleistocene Variations in the Earth's Orbital mantle plumes [Richards and Griffith, 1988]. Finally, as Parameters noted in the introduction, recent three-dimensional simulations of mantle convection [Zhanil and Yuen, 1995; In Figure 8 we show predictions of GIA-induced per- Bunlie et al., 1990] have shown that Earth-like plan- turbations in the Earth's precession constant H (H = forms (i.e., sheet-like downwellings) are obtained when (A+B)]/C where C is the polar moment of inertia the viscosity increasesignificantly (a factor of >_ 20) and A and B are the equatorial moments of inertia) relwith depth. ative to the predicted present-day perturbation (in fact, The inverted viscosity model (solid line, Figure we plot [SH(t)-SH(O)]/H)for the last 2.6 Myr. Details exhibits a rather intriguing jump at - 1000 km depth. of the calculation (for a given viscoelastic Earth model) may be tempting to ascribe some significance to this re- are given by Mitrovica and Forte [1995] and Mitrovica et suit, particularly in light of studies showing compelling al. [1996]. The ice history used in the calculation is conevidence for a seismic discontinuity (geodynamic bar- structed by combining load increments associated with rier?) at this depth [e.g., Kawakatsu and Niu, 1994]. the ICE-3G model of the final late Pleistocene deglacia- (Kawakatsu and Niu [1994] have argued for an S wave tion event [Tushingham and Peltier, 1992] in order to velocity contrast at 20 km depth which is roughly 40% generate a eustatic sea level fluctuation which matches of the contrast at the 070-kin discontinuity.) How- oxygen isotope ratios over this time period [Shackleton -2 MF2 FDW-GIA -2 I I I ' I I ' I ' 2800 2400 2000 1600 1200 800 400 0 Time (kyr B.P.) Figure 8. The predicted perturbation 5H/H over the last 2.6 Myr relative to the prediction at t = 0, computed using the viscosity models FDW-GIA and MF2 described in the text. The ice model used in the calculation is constrained to match the oxygen isotope record over this same time period [Shackleton e! al., 1990]. The ocean load is computed using a gravitationally self-consistent theory.

2762 MITROVICA AND FORTE: JOINT INVERSIONS FOR MANTLE VISCOSITY et al., 1990]. The results, which represent an anomaly at spherical harmonic degree 2, are sensitive to the chosen eustatic variation but are insensitive to the detailed geometry of the late Pleistocene ice complexes [Mitrovica and Forte, 1995; Mitrovica et al., 1996]. Laskar [1990] and Laskar et al. [1993] have developed theoretical/numerical techniques for accurately integrating the orbital elements of the Earth over long periods. Using these techniques, Laskar et al. [1993] concluded, on the basis of simple rigid-earth calculations, that perturbations in the precession constant arising from the Pleistocene ice mass fluctuations would be sufficient to induce a resonant response in the Earth's orbital parameters (mean precession frequency, obliquity, etc.) The resonance regime, arising from an external forcing associated with the perhelion of Jupiter and Saturn, would be entered for mean perturbations in the precession frequency ({[SH(t)-SH(O)]/H}, where angle brackets denote the mean) that are negative and larger in amplitude than 0.0015. The potential influence of this resonance is now considered to be the single largest uncertainty in orbital integrations extending through the Pleistocene [Laskar et al., 1993]. The work of Laskar et al. [1993] motivated studies of Mitrovica et al. [1994b] and Mitrovica and Forte [1995], who found that viscoelastic deformations of the planet (not considered by Laskar e! al. [1993])could significantly reduce the predicted perturbation to H. Using the time series in Figure 8, we have computed mean perturbations of-0.00067 and-0.00071 for viscosity models FDW-GIA and MF2, respectively. These predictions are a factor of 2 smaller than the resonance threshold and a factor of 3-4 smaller than an analogous calculation we have performed using a rigid Earth (which yielded a mean perturbation of-0.0024). We conclude that the GIA signal alone is likely insufficient to perturb the orbital system into the resonance regime described by Laskar et al. [1993]. In an orbital system not influenced by resonance, the mean perturbation in the precession frequency from some nominal value Po should be linearly proportional to the mean value of the time series [SH(t)- 5H(O)]/H (the proportionality constant is simply Po). To verify this, we have used codes made publically available by Laskar et al. [1993] to perform orbital integrations for the last 2.6 Myr. In the case when there is no perturbation to the precession constant, we have computed a nominal value of Po = 50.45 l"/yr. When the time series in Figure 8 are adopted, this nominal value is perturbed by-0.034"/yr and-0.036"/yr for models FDW-GIA and MF2, respectively, just as one would predict from linear theory. These perturbations refer to an increase in the precession period of about 20 years. The orbital integration software provided by Laskar e! al. [1993] also yields a time series of obliquity variations. In Figure 9 we show this time series for the nominal solution (SH = 0) and perturbations from this solution that arise from GIA-induced variations in the precession constant (Figure 8). The nominal time series oscillates around a value of 23.3 ø. The perturbation associated with the FDW-GIA and MF2 models grows as the integration proceeds backward from the present day, and this can be explained, to first order, by a small perturbation in the frequency of the obliquity variation. Since linear theory holds for these integrations, the perturbation in the frequency of the obliquity will be nearly equal to the perturbation in the frequency of preces- sion (i.e., -0.035"/yr). Over the 2.6 Myr integration, this perturbation introduces a time shift that reaches roughly 7% of a full obliquity cycle (of 41 kyr). This shift accounts for the rather significant perturbation at 2.6 Myr B.P. on the bottom two time series in Figure 9. Although difficult to discern, the perturbations to the nominal obliquity time series shown in Figure 9 also include a secular drift in the obliquity angle computed on the basis of results given by Ito et al. [1995]. The drift is due to a process termed "climatic friction" [Rubincam, 1990], which refers to a feedback loop associated with a coupling between ice sheet development and the orbital parameters. Using an "accepted" value for the time lag between insolation variations and ice sheet de- velopment (9 kyr; see Ito et al. [1995]), we predict a secular drift of 0.04ø/Myr when we adopt either of our new viscosity models (FDW-GIA or MF2). This drift amounts to 0.10 ø over the course of the 2.6 Myr integrations shown in Figure 9. This rate is near the maximum quoted by Ito e! al. [1995] in their viscosity model sensitivity analysis. Late Holocene Variations in the Length of Day Analysis of ancient eclipse records provides a means for constraining late Holocene variations in the length of day [e.g., Currot, 1966]. In particular, one can estimate the difference between the recorded time of oc- currence of an eclipse event and the predicted time of occurrence under the assumption that the Earth's ro- tation rate has remained fixed to its present value. We will call this time shift AT. A recent revised estimate of the time series AT(t), for a period extending to 2.5 kyr B.P. has been obtained by Stephenson and Morrison [1995], and their results are shown in Figure 10. The eclipse records indicate a significant deceleration of the Earth's rotation rate with time or an increase in the length of day, which is to be expected in the presence of tidal dissipation; an accurate estimate of the influence of tidal dissipation on length-of-day variations is shown by the upper shaded region on Figure 10 [after Stephenson and Morrison, 1995]. The discrepancy between the tidal dissipation prediction and the observed time shift indicates a secular, nontidal acceleration of the Earth's rate of rotation. This nontidal contribution has been associated, in past analyses of the same effect, with GIA [e.g., Sabadini and Peltlet, 1981; Wu and Peltlet, 1983]. In a calculation that is essentially identical to that which generated Figure 8, we have computed relative variations in the rotation rate due to GIA for our new viscosity model (FDW-GIA and MF2) over the last 3 kyr B.P. and converted these by integrating twice (once to obtain changes in the length of day and a second time to obtain the total time shift associated with these

MITROVICA AND FORTE- JOINT INVERSIONS FOR MANTLE VISCOSITY 2763 28 24-23-,I i.,.l,.,, I i II 22- vvvvvvvvvvvvv,vvv.vvv.,vvv,vv,vvvv,v,,v...,... I VvvvviVVVVVVVVVVV.VV,.,vvvvvvvvvvvvvvv,v... FDW-GIA 2700 2100 1500 900 300 Time (kyr B.P.) Figure 9. Obliquity of the Earth over the last 2.6 Myr (top solid line) for an orbital integration in which the precession constant is fixed to its observed present-day value. The remaining lines represent perturbations in the obliquity relative to this nominal solution for integrations which adopt the GIA-induced perturbations to.the precession constant shown in Figure 8. (The perturbations actually oscillate about 0.0; however, they are shifted by varying amounts for the purpose of display.) The perturbation time series also incorporate the secular drift in obliquity associated with "climatic friction."' In this case we adopt the results of Ito et al. [1995] and assume a lag time between ice sheet and insolation variations of 9 kyr. changes) to time series of AT(t). The results, which ice complexes over the last 2.5 kyr B.P.; our model ice incorporate the influence of tidal dissipation, are shown history assumes all melting is complete by 5 kyr B.P.) superimposed on Figure 10. must be small or, more accurately, must combine to con- Model FDW-GIA yields a net AT(t) time series that tribute a negligible signal. In contrast, the net AT(t) fits very well the constraints provided the timing of time series predicted using model MF2 falls somewhat eclipse events. In this case we conclude that other below the observed. range. In this case an additional potential signals in the AT(t) time series (e.g., ongo- signal would be required to bring the predictions into ing melting or growth of the Greenland or Antarctic accord with the observations. As an example, we have found that a moderate rate of net melting from the large polar ice sheets over the last 2.5 kyr, equivalent to a eu-,,, m I ''' i I! i m,,,,,,,,,,,,,...,... 500 1000 1500 YEAR (A.D.) Figure!0. The time shift AT between the recorded occurrence of ancient eclipse events and the predicted time of occurrence assuming a constant axial rotation rate equal to the present value [after Stephenson and Morrison, 1995]. The top shaded region (labeled TD) is a prediction of the range in AT(t) expected on the basis of tidal dissipation [also after Stephenson and Morrison, 1995]. The remaining shaded regions represent the total AT(t) computed by correcting the TD results for the predicted nontidal acceleration due to GIA on Earth models having the FDW-GIA, MF2, and PJ viscosity profiles (as defined in the text). The GIA predictions are based on the ice model used to generate the time series in Figure 8 and a gravitationally self-consistent ocean load.

ß 2764 MITROVICA AND FORTE: JOINT INVERSIONS FOR MANTLE VISCOSITY static sea level rise of 0.4 mm/yr, would be sufficient in vica and Peltier, 1993b; Trupin, 1993; James and Ivins, this regard. 1995]. At the lowest degrees the polar mass flux contribution to the j harmonics is relatively insensitive Present-Day Secular Variations in the Zonal to the geometry of the mass balance. Indeed, Mitrovica Harmonics of the Earth's Geopotential jt and Peltier [1993b] have shown that a total melting rate Predictions of harmonics, for spherical harmonic equivalento i mm/yr of sea level rise from the Greendegrees up to 7, are listed in Table 1. Results are land and Antarctic ice complexes will produce a signal given for both the FDW-GIA and MF2 viscosity mod- of j2 04 x 10-1!year-1, regardless of the pattern of els, and the calculations adopt the ice history speci- mass loss or even the partitioning of the mass balance fied above. Analyses of the perturbation of satellite between the two regions. (The J2 signal arising from orbits have provided constraints the j harmon- other levels of melting or growth can be computed by ics, at least at the lowest degrees [e.g., Yoder et al., scaling the above value.) Mitrovica and Peltlet [1993b] 1983; Rubincam, 1984; Cheng et al., 1989; Gegout and argued that once th.e radial viscosity profile of the man- Cazenave, 1993]. As an example, Eanes and Bettadpur tle (and hence the J2 signal due to GIA) was sufficiently [1996] used a 19-year time series of LAGEOS 1 observa- constrained, the observed value for J2 could be used to tions to derive the following constraint equation for the estimate the present-day rate of change of the total polow-degree zonal harmonics: J2 + 0.371J4 + 0.079J6 = lar ice mass. We follow up this suggestion using our (-2.64-0.3) x 10- year - (see also Eanes [1995]).(The new viscosity inferences. weightings that define this "effective" J2 are functions The observational constraint on J2, corrected for of the satellite orbit; an analogous estimate using Star- the melting of Meier's [1984] sources, has a value of lette data has a sensitivity that extends to significantly (-3.7 4-0.3) x 10-11year -1. Correcting this value for higher degrees [Eanes and Bettadpur, 1996].) Further- the GIA signal based on the FDW-GIA and MF2 viscosmore, using a multisatellite laser ranging (SLR) so- ity models (Table 1) yields (-0.7 4-0.3) x 10-Xlyear -1 lution, Netera and Klosko [1996] have estimated that and (0.94-0.3) x 10-1Xyear -1, respectively. In the case J2 - (-2.774-0.2.5) x 10 - year -. The observed J signal may have contributions from a variety of signals, most notably, ongoing melting of small ice sheets and glaciers [Meier, 1984] and a net mass flux between the large polar ice caps (Antarctic and Greenland) and the global oceans. The melting events tabulated by Meier [1984] produce a eustatic sea level rise of 0.4 mm/yr and contribute a signal of 10- year- to J2 and less than 10-2year to most other harmonics [e.g., $abadini et al., 1988; Mitrovica and Peltier, 1993b]. The present-day mass balances of the large polar ice caps are uncertain, even in their sign, and so too are their contributions to secular variations in the geopotential harmonics. A number of authors have, nevertheless, predicted this contribution on the basis of various scenarios for the recent polar ice history [e.g., Sabadini et al., 1988; Ivins et al., 1993; Mitro- Table 1. Inverted Model Predictions Observable Model Model FDW-GIA MF2-3.0-4.6 0.9 1.4-3.2-3.8 3.0 3.4-1.15-1.8 2.2 3.2-4.3-6.1 All predictions are in units of 10-11 year -1. Jt refers to the secular variation of the degree t zonal harmonic of the geopotentim, and Jt = J +0.371J4+0.07 4. See text for model descriptions. of the FDW-GIA model this implies a polar ice sheet growth event equivalent to a eustatic sea level fall of 0.1-0.25 mm/yr or a total eustatic sea level rise (after adding 0.4 mm/yr from Meier's [1984] sources) of 0.15-0.3 mm/yr. Performing the same budget calculation for the case of the MF2 viscosity model yields a polar ice cap melt event equivalento 0.15-0.3 mm/yr and a total eustatic sea level rise of 0.55-0.7 mm/yr. These contraints are somewhat misleading since there is a possibility that other geophysical processes (e.g., mantle deformation driven by pressure variations at the CMB [Fang et al., 1996]) may give rise to a non-. negligible J2 signal. Nevertheless, if the range is ap- proximately correct, then it would imply that a significant portion of the ongoing global sea level rise of 1.5-2.0 mm/yr inferred from tide gauge records [e.g., Peltier and Tushingham, 1989; Douglas, 1991; Davis and Mitrovica, 1996] likely originates from the steric effect of ocean thermal expansion. We can repeat the above budget calculation using the LAGEOS i constraint on the lumped variation J2 +0.371J4 +0.079J6 [Eanes, 1995; Eanes and Bettad- put, 1996]. The resulting upper bound on the mass flux from the large polar ice complexes to the global oceans, 0.6 mm/yr, is less robust since the predicted GIA signal in the higher-degree harmonics becomes quite sensitive to the detailed geometry of the late Pleistocene ice sheets. Stephenson and Morrison [1995] have pointed out that their constraint on late Holocene length-of-day variations is "consistent" with modern, satellite-derived estimates of J2. That is, a GIA model prediction that fits one of these data types should also fit the other. Model FDW-GIA is an obvious example of this consistency (see Figure 10 and Table 1). The results for this model imply that the net signal from non-gia sources

ß. ß MITROVICA AND FORTE: JOINT INVERSIONS FOR MANTLE VISCOSITY 2765 (e.g., ongoing melting events, melting events extending over the last few thousand years, etc.) is relatively minor. Furthermore, in the case of model MF2, the observed late Holocene length-of-day variation and the present-day 2 signal can both be reconciled by invoking a moderate amount of mass flux from the large polar ice complexes (equivalento a eustatic sea level rise of - 0.3-0.4 mm/yr) over the last 2.5 kyr. Final Remarks Our analyses have suggested that data associated with mantle convection and GIA may be reconciled with a single viscosity profile having a detailed depth dependence (Figure 6). In a recent article, Peltier and Jiang [1996, p. 3269] have suggested that a much simpler model, characterized by a constant viscosity of 1021 Pa s down to 1400 km depth and 14 x 1021 Pa s below this boundary, affects "a reconciliation between the requirements of these distinct geophysical observations." This conclusion was based on a suite of forward calculations involving polar wander and J2 variations due to GIA. The suggestion that convection data were simultaneously fit by such a model was not supported by quantitative analysis. It is important, given the effort involved in the present study, to understand whether much simpler models of the type advocated by Peltlet and Jiang [1996] can, in fact, yield a reconciliation. ß J2 results presented and Klosko, 1996]. ß The argument made by Peltier and Jiang [1996] is that this constraint is fit either by a fac- tor of - 2 increase across 670 km depth or a factor of - 14 increase across 1400 km depth. The latter model (hereinafter referred to as PJ) is accepted by them on the basis of the general requirements of convection observables for a viscosity increase with depth. The Peltlet and Jiang [1996] predictions are incorrect. The two sets of viscosity models they consider converge to the same model (an isoviscous 1021 Pa s mantle) when the viscosity of the deeper layer is 1021 Pa s. However, their predictions clearly do not converge to the same result as the abscissa trends toward the isoviscous model (see Figure 11a). Indeed, according to Peltlet and Jiang [1996], the J2 signal predicted for the isoviscous model represented by the leftmost value of the solid line in Figure 11a is the same as the signal associated with a model in which the viscosity jumps an order of magnitude, to 1022 Pa s, below 1400 km 1o I I I I I Illl I I I I I Illl I -. i... :::::::: I I I I I IIII I I I I I IIII I 21 22 23 10 1o 10 Deep Mantle Viscosity (Pa s) by Peltier and Jiang [1996] are replotted in Figure 11a. The solid line in Figure 11a rep- Figure 11. (a) Predictions of present-day J2 variaresents a suite of predictions in which the upper mantle tions due to GIA presented by Peltier and Jiang [1996, viscosity (i.e., above 670 km depth) is fixed to 1021 Pa s Figures 8b and 9b]. All results were computed using a and the (constant) lower mantle viscosity was varied 2 lithospheric thickness of 120 km and a two-layer mantle viscosity profile in which the upper layer is fixed to orders of magnitude. The dotted line is generated in a 1021 Pa s. The lower layer has a viscosity that is varied similar fashion, with the exception that the boundary according to the abscissa. The solid and dotted lines between the two isoviscous layers is placed at 1400 km refer to results generated by placing the boundary bedepth, and the vlscosity below this region is varied. tween the two layers at 670 and 1400 km depth, respec- These results are given by Peltier and Jiang [1996, Fig- tively. The shaded region represents the observational ures 8b and 9b]. The observational constraint on J2 constraint [Nerem and Klosko, 1996]. (b) Our own calis shown by the shaded region in Figure 11a [Nerem culation of J. variations for the same set of Earth mod- els considered in Figure 11a. The surface load we adopt is the same as that used to generate Figure 8. depth. (We note that predictions of the present-day rate of polar wander presented by Peltlet and Jiang [1996, Figures 7c and 9a] are clearly incorrect for the same reason.) The origin Of this error is uncertain, but the implications are dra-matic. To investigate this, we have repea ed the Peltier and Jiang [1996] calculations using the ice history described in the context of Figure 8; the results are shown in Figure 11b. The solid line in Figure 11b (constructed by varying a constant lower mantle viscosity) consistent with the results of both Ivins et al. [1993] and Mitrovica and Peltier [1993b]. The dotted line is essentially the same as that presented by Mitrovica and Peltier [1993b; Figure 3], who considered the case of a two-layer viscosity model with a boundary at 1200 km depth. Note that the two sets of calcula-

2766 MITROVICA AND FORTE: JOINT INVERSIONS FOR MANTLE VISCOSITY tions in Figure lib converge at the isoviscous 102 Pa s an NSERC research grant, and A.M.F. acknowledges the mantle model. Furthermore, the two suites of models support of the IPGP (URA CNRS 195). yield similar J2 predictions in the case of deep mantle viscosities below 30 x 102. As discussed by Mitrovica and Peltlet [1993b], the J2 datum is generally sensitive References to variations in viscosity in the deepest region of the mantle; hence placing the viscosity jump at 670 or 1400 Andrews, J. T., A Geomorphological Study oj Postglacial km depth makes little difference to the predictions un- Uplift With Particular Re erence to Arctic Canada, Oxtil the deep mantle viscosity is high enough that the ford Univ. Press, New York, 1970. Andrews, J. T., and L. Drapier, Radiocarbon dates obta/ned sensitivity has migrat. ed to shallower depths. through Geographical Branch field observations, Geogr. Clearly, the GIA J2 signal (correctly) predicted us- Bull., 9, 115-162, 1967. ing the PJ viscosity model (-5.5 x 10- year -, Figure Bunge, H.-P., M. A. Richards, and J. R. Baumgardner, Ef- 11b) cannot alone reconcile the observe datum. The fect of depth-dependent viscosity on the planform of mansame is true of a prediction of the late Holocene time tle convection, Nature, 379, 436-438, 1996. shift (Figure 10). In both cases we would be forced Cathles, L. M., The Viscosity oj the Earth's Mantle, Princeton Univ. Press, Princeton, N.J., 1975. to seek contributions from other non-gia sources (e.g., Cheng, M., R. Eanes, C. Shum, B. Schutz, and B. Tapley, ongoing ice melting equivalento -01 mm/yr would be Temporal variations in low degree zonal harmonics from required to satisfy the 0/2 constraint). In addition, we Starlette orbit analysis, Geophys. Res. Lett., 16, 393-396, have found that the PJ model does not yield an ac- 1989. ceptable fit to the long-wavelength, free-air gravity har- Chopelas, A., Sound velocities of MgO to very high commonics. In the case of the density heterogeneity model pression, Earth Planet. Sci. Lett., 11 /, 185-192, 1992. S.F1.K and the velocity-to-density conversion presented Constable, S.C., R. L. Parker, and C. G. Constable, Occam's inversion: A practical algorithm for generating by Karato [1993], the PJ model increases the variance smooth models from electromagnetic sounding data, Geoof the observations by a factor of 1.5. (The variance increases by a factor of 0.7 when a constant velocity-todensity scaling of 0.2 is adopted.) We have shown that a single profile of mantle viscosity can satisfy independent constraints associated with long-wavelength, free air gravity anomalies and post- glacial decay times. This results weakens (but does not rule out) the suggestion that transient effects may be important across timescales that bridge GIA and mantle convection (the first demonstration of this was by Forte and Mitrovica [1996]). Models FDW-GIA and MF2 are characterized by a rather detailed structure, which is necessary to obtain a simultaneous reconcil- Douglas, B.C., Global sea level rise, J. Geophys. Res., 96, iation of data subsets (uplift curves in Fennoscandia, 6981-6992, 1991. Hudson Bay, and Arctic Canada and long-wavelength, Dziewonski, A.M., and D. L. Anderson, Preliminary referfree-air gravity anomalies) with widely varying resolving ence Earth model (PREM), Phys. Earth Planet. Inter., powers. This structure also clearly plays an important 25, 297-356, 1981. Eanes, R. J., A study of temporal variations in the Earth's role in our a posterjori applications of the new models. gravitationm field using Lageos-1 laser range observations, For example, the model FDW-GIA is characterized by a stiff deep mantle region; nevertheless, a prediction of late Holocene length-of-day variations based upon it are able to satisfy the observational constraints (Figure 10) because the model is also characterized by a shallower region of low viscosity. In future work we will further constrain the radial profile of mantle viscosity by incorporating a number of other data types into our in- versions (e.g., a revised Fennoscandian relaxation spectrum, differential late Holocene sea level highstands at far-field sites, etc.) and by continuing our ongoing effort to increase the database of useful postglacial decay times. Acknowledgments. We thank G. Spada, D. A. Yuen, and an anonymous referee for their constructive reviews of the original manuscript. We acknowledge the generosity of J. Laskar for making his orbital integration codes available to the community. The work of J.X.M. was supported by physics, 52, 289-300, 1987. Corrieu, V., Y. Ricard, and C. Froidevaux, Converting mantle tomography into mass anomalies to predict the Earth's radial viscosity, Phys. Earth Planet. Inter., &i, 3-13, 1994. Currot, D. R., Earth deceleration from ancient solar echpses, Astron. J., 71, 264-269, 1966. Davis, J. L., and J. X. Mitrovica, Glacial isostatic adjustment and the anomalous tide gauge record of eastern North America, Nature, 379, 331-333, 1996. Dehant, V., M.-F. Loutre, and A. Berger, Potential impact of the northern hemisphere quaternary ice sheets on the frequencies of the astrochmatic orbital parameters, J. Geophys. Res., 95, 7573-7578, 1990. Ph.D. thesis, Univ. of Tex. at Austin, 1995. Eanes, R. J., and S. V. Bettadpur, Temporal variabihty of Earth's gravitational field from satellite laser. ranging, in Global Gravity Field and its Variations, Int. Assoc. of Geod. Symp. Set., edited by R. Rapp, in press, 1996. Fang, M., and B. H. Hager, A singularity free approach to post glacial rebound calculations, Geophys. Res. Lett., 21, 2131-2134, 1994. Fang, M., and B. H. Hager, The singularity mystery associated with a radially continuous Maxwell viscoelastic structure, Geophys. J. Int., 123, 849-865, 1995. Fang, M., B. H. Hager, and T. A. Herring, Surface deforma- tion caused by pressure changes in the fluid core, Geophys. Res. Lett., 23, 1493-1496, 1996. Fjeldskaar, W., and L. M. Cathles, The present rate of uplift of Fennoscandia imphes a low-viscosity asthenosphere, Terra Nova, 3, 393-400, 1991. Forte, A.M., and J. X. Mitrovica, A new inference of mantle viscosity based on a joint inversion of post-glacial rebound data and long-wavelength geoid anomalies, Geophys. Res. Lett., 23, 1147-1150, 1996.

MITROVICA AND FORTE: JOINT INVERSIONS FOR MANTLE VISCOSITY 2767 Forte, A.M., and W. R. Peltier, Plate tectonics and aspherical Earth structure: The importance of poloidal-toroidal coupling, J. Geophys. Res., 92, 3645-3679, 1987. Forte, A.M., and W. R. Peltier, Viscous flow models of global geophysical observables, 1, Forward problems, J. Geophys. Res., 96, 20,131-20,159, 1991. Forte, A.M., W. R. Peltier, and A.M. Dziewonski, Inferences of mantle viscosity from tectonic plate velocities, Geophys. Res. Lett., 18, 1747-1750, 1991. Forte, A.M., A.M. Dziewonski, and R. L. Woodward, Aspherical structure of the mantle, tectonic plate motions, nonhydrostatic geoid, and topography of the core-mantle boundary, in Dynamics of the Earth's Deep Interior and Earth Rotation, Geophys. Monogr. Set., vol. 72, edited by J.-L. Le MouS1, D. E. Smylie, and T. Herring, pp. 135-166, AGU, Washington, D.C., 1993. Forte, A.M., R. L. Woodward, and A.M. Dziewonski, Joint inversions of seismic and geodynamic data for models of three-dimensional mantle heterogeneity, J. Geophys. Res., 99, 21,857-21,877, 1994. Gasperini, P., and R. Sabadini, Lateral heterogeneitic in mantle viscosity and post-glacial rebound, Geophys. J. Int., 98, 413-428, 1989. Gegout, P., and A. Cazenave, Temporal variations in Earth's gravitational field for 1985-1989 derived from Lageos, Geophys. J. Int., 11 1, 347-359, 1993. Giunchi, C., G. Spada and R. Sabadini, Lateral viscosity variations and post-glacial rebound: effects on presentday VLBI baseline deformations, Geophys. Res. Lett., in press, 1996. Gurnis, M., Rapid continental subsidence following the initiation and evolution of subduction, Science, 255, 1556-1558, 1992. Gurnis, M., and G. F. Davies, Numerical study of high Rayleigh number convection in a medium with depth dependent viscosity, Geophys. J. R. Astron. Soc., 85, 523-541, 1986. Hager, B. H., Subducted slabs and the geoid: Constraints on mantle rheology and flow, J. Geophys. Res., 89, 6003-6015, 1984. Hager, B. H., Mantle viscosity: A comparison of models from post-glacial rebound and from the geoid, plate driving forces, and advected heat flux, in Glacial Isostasy, Sea-Level and Mantle Rheology, NATO ASI Set. C, vol. 334, edited by R. Sabadini, K. Lambeck, and E. Boschi, pp. 493-514, Kluwer Acad., Norwell, Mass., 1991. Hager, B. H., and R. W. Clayton, Constraints on the structure of mantle convection using seismic observations, flow models, and the geoid, in Mantle Convection: Plate Tectonics and Global Dynamics, edited by W.R. Peltier, pp. 657-763, Gordon and Breach, Newark, N.J., 1989. Hah, D., and J. Wahr, The viscoelastic relaxation of a realistically stratified Earth, and a further analysis of postglacial rebound, Geophys. J. Int., 120, 287-311, 1995. Hanyk, L., J. Moser, D. A. Yuen, and C. Matyska, Time- domain approach for the transient responses in stratified viscoelastic Earth models, Geophys. Res. Lett., œœ, 1285-1288, 1995. Haskell, N. A., The motion of a fluid under a surface load, 1, Physics, 6, 265-269, 1935. Haskell, N. A., The motion of a fluid under a surface load, 2, Physics, 7, 56-61, 1936. Ito, T., K. Masuda, Y. Hamano, and T. Matsui, Climatic friction: A possible cause for secular drift of the Earth's obliquity, J. Geophys. Res., 100, 15,147-15,162, 1995. Ivins, E. R., C. G. Sammis, and C. F. Yoder, Deep mantle viscous structure with prior estimate and satellite constraint, J. Geophys. Res., 98, 4579-4609, 1993. James, T. S., and E. R. Ivins, Present-day Antarctic mass changes and crustal motion, Geophys. Res. Lett., 22, 973-976, 1995. James, T. S., and A. Lambert, A comparison of VLBI data with the ICE-3G glacial rebound model, Geophys. Res. Lett., 20, 871-874, 1993. Johnston, P., The effect of spatially non-uniform water loads on predictions of sea level change, Geophys. J. Int., 11 1, 615-634, 1993. Karato, S.-I., Importance of anelasticity in the interpretation of seismic tomography, Geophys. Res. Lett., 20, 1523-1626, 1993. Kawakatsu, H., and F. Niu, Seismic evidence for a 920-km disco,,tinuity in the mantle, Nature, 371, 301-305, 1994. King, S. D., Radial models of mantle viscosity: Results from a genetic algorithm, Geophys. J. Int., 122, 725-734, 1995. King, S. D., and T. G. Masters, An inversion for the radial viscosity structure using seismic tomography, Geophys. Res. Lett., 19, 1551-1554, 1992. Lambeck, K., P. Johnston, and M. Nakada, Holocene glacial rebound and sea-level change in NW Europe, Geophys. J. Int., 103, 451-468, 1990. Laskar, J., The chaotic motion of the solar system: A numerical estimate of the size of the chaotic zones, Icarus, 88, 266-291, 1990. Laskar, J., F. Joutel, and F. Boudin, Orbital, precessional, and insolation quantities for the Earth from-20 Myr to +10 Myr, Astron. Astrophys., 270, 522-533, 1993. Manga, M., and R. J. O'Connell, The tectosphere and postglacial rebound, Geophys. Res. Lett., 22, 1949-1952, 1995. McConnell, R. K., Viscosity of the mantle from relaxation time spectra of isostatic adjustment, J. Geophys. Res., 73, 7089-7105, 1968. McKenzie, D. P., The viscosity of the mantle, Geophys. J. R. Astron. Soc., 1 1, 297-305, 1967. Meier, M. F., Contribution of small glaciers to global sea level, Science, 226, 1418-1421, 1984. Mitrovica, J. X., Haskell [1935] revisited, J. Geophys. Res., 101,555-569, 1996. Mitrovica, J. X., and A.M. Forte, Pleistocene glaciation and the Earth's precession constant, Geophys. J. Int., 121, 21-32, 1995. Mitrovica, J. X., and W. R. Peltier, Pleistocene deglaciation and the global gravity field, J. Geophys. Res., 9 1, 13,651-13,671, 1989. Mitrovica, J. X., and W. R. Peltier, A complete formalism for the inversion of postglacial rebound data: Resolving power analysis, Geophys. J. Int., 10 1, 267-288, 1991a. Mitrovica, J. X., and W. R. Peltier, On postglacial geoid subsidence over the equatorial oceans, J. Geophys. Res., 96, 20,053-20,071, 1991b. Mitrovica, Jo X., and W. R. Peltier, The inference of mantle viscosity from an inversion of the Fennoscandian relaxation spectrum, Geophys. J. Int., 11 1, 45-62, 1993a. Mitrovica, J. X., and W. R. Peltier, Present-day secular variations in the zonal harmonics of the Earth's geopotential, J. Geophys. Res., 98, 4509-4526, 1993b. Mitrovica, J. X., and W. R. Peltier, Constraints on mantle viscosity based upon the inversion of post-glacial uplift data from the Hudson Bay region, Geophys. J. Int., 122, 353-377, 1995. Mitrovica, J. X., J. L. Davis, and I. I. Shapiro, Constraining proposed combinations of ice history and Earth theol- ogy using VLBI-determined baseline length rates in North America, Geophys. Res. Lett., 20, 2387-2390, 1993. Mitrovica, J. X., J. L. Davis, and I. I. Shapiro, A spectral formalism for computing three-dimensional deformations due to surface loads, 2, Present-day glacial isostatic adjustment, J. Geophys. Res., 99, 7075-7101, 1994a.

2768 MITROVICA AND FORTE: JOINT INVERSIONS FOR MANTLE VISCOSITY Mitrovica, J. X., R. Pan, and A.M. Forte, Late Pleistocene and Holocene ice mass fluctuations and the Earth's precession constant, Earth. Planet. Sci. Lett., 128, 489-500, 1994b. Mitrovica, J. X., A.M. Forte, and R. Pan, Glaciation induced variations in the Earth's precession frequency, obhquity and insolation over the last 2.6 Ma, Geophys. J. Int., in press, 1996. Nakada, M., and K. Lambeck, Glacial rebound and relative sea level variations: A new appraisal, Geophys. J. R. Astron. Soc., 90, 171-224, 1987. Nakada, M., and K. Lambeck, Late Pleistocene and Holocene sea-level change in the Australian region and mantle theology, Geophys. J. Int., 96, 497-517, 1989. Nakiboglu, S. M., and K. Lambeck, Deglaciation effects upon the rotation of the Earth, Geophys. J. R. Astron. Soc., 62, 49-58, 1980. Nerem, R. S., and S. M. Klosko, Secular variations of the zonal harmonics and polar motion as geophysical constraints, in Global Gravity Field and its Variations, Int. Assoc. of Geod. Syrnp. Set., edited by R. Rapp, in press, 1996. O'Connell, R. J., Pleistocene glaciation and the viscosity of the lower mantle, Geophys. J. R. A stroh. Soc., 23, 299-327, 1971. Parsons, B. D., Changes in the Earth's shape, Ph.D. thesis, Cambridge Univ., Cambridge, England, 1972. Peltier, W. R., The impulge response of a Maxwell earth, Rev. Geophys., 12, 649-669, 1974. Peltlet, W. R., New constraint on transient lower mantle rheology and internal mantle buoyancy from glacial rebound data, Nature, 318, 614-617, 1985. Peltier, W. R., Ice age paleotopography, Science, 265, 195-201, 1994. Peltlet, W. R., and J. T. Andrews, Glacial isostatic adjustment, I, The forward problem, Geophys. J. R. Astron. Soc.,,6, 605-646, 1976. Peltier, W. R., and X. Jiang, The procession constant of the Earth: Variations through the ice age, Geophys. Res. Left., 21, 2299-2302, 1994. Peltier, W. R., and X. Jiang, Glacial isostatic adjustment and Earth rotation: Refined constraints on the viscosity of the deepest mantle, J. Geophys. Res., 101, 3269-3290, 1996. Peltier, W. R., and A.M. Tushingham, Global sea level rise and the greenhouseffect: Might they be connected?, Ont., 1989. Science, 2, 806-810, 1989. Tushingham, A.M., and W. R. Peltier, ICE-3G: A new Peltier, W. R., R. A. Drummond, and A.M. Tushingham, Post-glacial rebound and transient lower mantle theology, Geophys. J. R. Astron. Soc., 87, 79-116, 1986. global model of late Pleistocene deglaciation based upon geophysical predictions of postglacial relative sea level change, J. Geophys. Res., 96, 4497-4523, 1991. Ricard, Y., and C. Vigny, Mantle dynamics with induced plate tectonics, J. Geophys. Res., 9J, 17,543-17,559, 1989. Tushingham, A.M., and W. R. Peltier, Vahdation of the Ricard, Y., L. Fleitout, and C. Froidevaux, Geoid heights and hthospheric stresses for a dynamic Earth, Ann. Geophys., 2, 267-286, 1984. Ricard, Y., C. Vigny, and C. Froidevaux, Mantle heterogeneities, geoid and plate motion: A Monte Carlo inversion, J. Geophys. Res., 9J, 13,739-13,754, 1989. Richards, M. A., Hot spots and the case for a high viscosity lower mantle, in Glacial Isostasy, Sea-Level and Mantle Rheology, NATO ASI Set. C, vol. 334, edited by R. Sabadini, K. Lambeck, and E. Boschi, pp. 571-588, Kluwer Acad., Norwell, Mass., 1991. Richards, M. A., and R. W. Grifiiths, Deflection of plumes by mantle shear flow: Experimental results and a simple theory, Geophys. J. Int., 9J, 367-376, 1988. Richards, M. A., and B. H. Hager, Geoid anomalies in a dynamic Earth, J. Geophys. Res., 89, 5987-6002, 1984. Rubincam, D. P., Post glacial rebound observed by Lageos and the effective viscosity of the lower mantle, J. Geophys. Res., 89, 1077-1087, 1984. Rubincam, D. P., Mars: Change in axial tilt due to chmate?, Science, 2 8, 720-721, 1990. Sabadini, R., and W. R. Peltier, Pleistocene deglaciation and the Earth's rotation: Imphcations for mantle viscosity, Geophys. J. R. Astron. Soc., 66, 553-578, 1981. Sabadini, R., and D. A. Yuen, Mantle stratification and long-term polar wander, Nature, 339, 373-375, 1989. Sabadini, R., D. A. Yuen, and P. Gasperini, The effects of transient theology on the interpretation of lower mantle rheology, Geophys. Res. Lett., 12, 361-365, 1985. Sabadini, R., D. A. Yuen, and M. Porthey, The effects of upper-mantle lateral heterogeneities on post-glacial rebound, Geophys. Res. Lett., 13, 337-340, 1986. Sabadini, R., D. A. Yuen, and P. GasperJul, Mantle theology and satellite signatures from present-day glacial forcings, J. Geophys. Res., 93, 437-447, 1988. Shackleton, N.J., A. Berger, and W. R. Peltier, An alternative astronomical calibration of the lower Pleistocene timescale based on ODP site 677, Trans. R. Soc. Edinburgh Earth Sci., 81, 251-261, 1990. Spada, G., Y. Ricard, and R. Sabadini, Excitation of true polar wander by subduction, Nature, 360, 452-454, 1992. Steinberger, B. M., Motion of hotspots and changes of the Earth's rotation axis caused by a conveering mantle, Ph.D. thesis, Harvard Univ., Cambridge, Mass., 1996. Stephenson, F. R., and L. V. Morrison, Long-term fluctuations in the Earth's rotation: 700 BC to 1990 AD, Philos. Trans. R. Soc. London Set. A., 351, 165-202, 1995. Trupin, A. S., Effect of polar ice on the Earth's rotation and gravitational potential, Geophys. J. Int., 113, 273-283, 1993. Trupin, A. S., and J. M. Wahr, Constraints on long-period sea level variations from global tide gauge data, in Glacial Isostasy, Sea-Level and Mantle Rheology, NATO ASI Set. C, vol. 334, edited by R. Sabadini, K. Lambeck, and E. Boschi, pp. 271-284, Kluwer Acad., Norwell, Mass., 1991. Trupin, A. S., M. F. Meier, and J. M. Wahr, Effect of melting glaciers on the Earth's rotation and gravitational field: 1965-1984, Geophys. J. Int., 108, 1-15, 1992. Tushingham, A.M., A global model of late Pleistocene deglaciation: Imphcations for Earth structure and sea level change, Ph.D. thesis, Univ. of Toronto, Toronto, ICE-3G model of Wurm-Wisconsin deglaciation using a global data base of relative sea level histories, J. Geophys. Res., 97, 3285-3304, 1992. Wolf, D., An upper bound on hthosphere thickness from glacio-isostatic adjustment in Fennoscandia, J. Geophys., 61, 141-149, 1987. Wolf, D., Notes on estimates of the glacial-isostatic decay spectrum for Fennoscandia, Geophys. J. Int., in press, 1996. Woodward, R. L., A.M. Forte, W.-J. Su, and A.M. Dziewonski, Constraints on the large-scale structure of the Earth's mantle, in Evolution of the Earth and Planets, _Geophys. Monogr. Set., vol. 7, edited by E. Takahashi, R. Jeanloz, and D.C. Ruble, pp. 89-109, AGU, Washington, D.C., 1993. Wu, P., and W. R. Peltlet, Glacial isostatic adjustment and the free air gravity anomaly as a constraint on deep mantle viscosity, Geophys. J. R. A stron. Soc., 7, 377-450, 1983.

MITROVICA AND FORTE: JOINT INVERSIONS FOR MANTLE VISCOSITY 2769 Wu, P., and W. R. Peltier, Pleistocene deglaciation and the Earth's rotation: A new analysis, Geophys. J. R. Astron. $oc., 76, 753-791, 1984. Yoder, C., J. Williams, J. Dickey, B. Schutz, R. Eanes, and B. Tapley, Secular variations of the Earth's gravitational harmonic J2 coefficient from Lageos and non-tidal accel- eration of Earth rotation, Nature, $05, 757-759, 1983. Yuen, D. A., R. Sabadini, and E. V. Boschi, Viscosity of the lower mantle as inferred from rotational data, J. Geophys. Res., 87, 10,745-10,762, 1982. Yuen, D. A., R. Sabadini, P. Gasperini, and E. V. Boschi, On transient rheology and glacial isostasy, J. Geophys. Res., 91, 11,420-11,438, 1986. Zhang, S., and D. A. Yuen, The influences of lower mantle viscosity stratification on 3D spherical-shell mantle convection, Earth Planet. Sci. Lett., 132, 157-166, 1995. A.M. Forte, D4partement de Sismologie, Institut de Physique du Globe de Paris, Paris, France. J. X. Mitrovica, Department of Physics, University of Toronto, 60 St. George Street, Toronto, Ontario, Canada MSS 1A7. (e-mail: jxm@physics.utoronto.ca) (Received March 18, 1996; revised September 30, 1996; accepted October 11, 1996.)