Present-day secular variations in the low-degree harmonics of the geopotential: Sensitivity analysis on spherically symmetric Earth models

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 107, NO. B12, 2378, doi:10.1029/2001jb000696, 2002 Present-day secular variations in the low-degree harmonics of the geopotential: Sensitivity analysis on spherically symmetric Earth models M. E. Tamisiea and J. X. Mitrovica Department of Physics, University of Toronto, Toronto, Ontario, Canada J. Tromp Seismological Laboratory, California Institute of Technology, Pasadena, California, USA G. A. Milne Department of Geological Sciences, University of Durham, Durham, UK Received 24 May 2001; revised 30 May 2002; accepted 18 July 2002; published 31 December 2002. [1] We predict present-day rates of change of the long-wavelength components of the Earth s geopotential due to the late Pleistocene glacial cycles (hereinafter referred to as glacial isostatic adjustment, or GIA). These predictions are generated using spherically symmetric, self-gravitating, (Maxwell) viscoelastic Earth models. Previous studies have generally considered only zonal (i.e., azimuthally independent) harmonics, or the so-called _J coefficients. We extend these efforts to focus on the nonzonal harmonics and explore the sensitivity of our predictions to changes in a variety of model inputs. As an example,we examine the influence on the GIA predictions of ignoring rotational perturbations and assuming an elastically incompressible model and find that these assumptions can have a significant impact on the degree-2, order-1 coefficients, while the other harmonics are generally affected by <10%. Though the GIA predictions are generally insensitive to variations in lithospheric thickness and upper mantle viscosity, reasonable variations in either lower mantle viscosity or the late Pleistocene ice history will have a more significant effect. Mindful of upcoming dedicated gravity missions, we show that predictions of the GIA-induced changes of the geoid can vary by at least 0.5 mm yr 1 over a plausible range of Earth and ice models. INDEX TERMS: 1213 Geodesy and Gravity: Earth s interior dynamics (8115, 8120); 1236 Geodesy and Gravity: Rheology of the lithosphere and mantle (8160); 8162 Tectonophysics: Evolution of the Earth: Rheology mantle; KEYWORDS: secular, geoid, GIA, Stokes coefficients Citation: Tamisiea, M. E., J. X. Mitrovica, J. Tromp, and G. A. Milne, Present-day secular variations in the low-degree harmonics of the geopotential: Sensitivity analysis on spherically symmetric Earth models, J. Geophys. Res., 107(B12), 2378, doi:10.1029/ 2001JB000696, 2002. 1. Introduction [2] Most previous studies of the present-day secular change of the Earth s geopotential (or geoid height) have focused on the low-degree zonal coefficients of its spherical harmonic expansion, with particular emphasis on the degree two case (i.e., _J 2 ). This focus reflects both the limited nature of estimates available from previous gravity missions [e.g., Yoder et al., 1983; Eanes and Bettadpur, 1996; Cazenave et al., 1996; Nerem and Klosko, 1996; Cheng et al., 1997; Klosko and Chao, 1998], and the mathematical connection between load-induced _J 2 and length-of-day variations [e.g., Wu and Peltier, 1984; Mitrovica and Milne, 1998]. Copyright 2002 by the American Geophysical Union. 0148-0227/02/2001JB000696$09.00 [3] Glacial isostatic adjustment (GIA) refers to the response of the Earth system to loading associated with the late Pleistocene glacial cycles. The contribution of GIA to secular variations in the zonal harmonics of the Earth s geopotential has been the focus of nearly two decades of study [e.g., Peltier, 1983; Wu and Peltier, 1984; Yuen and Sabadini, 1985; Ivins et al., 1993; Mitrovica and Peltier, 1993; Vermeersen et al., 1998; Devoti et al., 2001]. Several analyses have revealed a detailed depth-dependent sensitivity of numerical predictions to variations in the radial profile of mantle viscosity [e.g., Mitrovica and Peltier, 1993; Devoti et al., 2001]; in the case of the lowest-degree harmonics this sensitivity extends into the deepest regions of the mantle. The sensitivity of the predictions to variations in the ice load has also been explored. Wu and Peltier [1984], for example, showed that the GIA-induced _J 2 signal was primarily sensitive to net melting of late Pleistocene ice sheets (or the ETG 18-1

ETG 18-2 TAMISIEA ET AL.: SENSITIVITY OF SECULAR GEOID PREDICTIONS equivalent eustatic sea level change associated with the deglaciation). Mitrovica and Peltier [1993] and Ivins et al. [1993] extended this sensitivity analysis to higher degrees and noted a strong sensitivity of predicted odd-degree zonal harmonics (particularly at degree 3) to variations in the ice model. In contrast to degree two, the sign of the Legendre polynomial basis function at degree three is different at the two poles of the Earth. As a consequence, the _J 3 prediction is sensitive to small changes in the balance between mass loss from northern and southern ice complexes. [4] The impact of present-day glacial forcings on the zonal harmonics has also been an area of active interest [e.g., Yoder and Ivins, 1985; Sabadini et al., 1988; Mitrovica and Peltier, 1993; James and Ivins, 1997]. In this case, the Earth s response is primarily elastic and hence uncertainties in the radial viscosity profile are largely irrelevant. At degree two the predictions are sensitive to the net ongoing mass flux from the global ice reservoirs, and thus an accurate prediction of this signal can be obtained without knowledge of the individual contributions to the net flux [Mitrovica and Peltier, 1993]. For the reason discussed above, this simplified requirement does not hold for odd degree zonal harmonics [Mitrovica and Peltier, 1993; James and Ivins, 1997]. [5] Nonzonal secular variations in the geopotential, induced by either GIA or ongoing ice mass changes, have received less attention. Yuen et al. [1987] predicted zonal and nonzonal harmonics for a range of different lower mantle viscosities and both Maxwell and Burger s body rheologies. Their GIA calculations did not vary the upper mantle viscosity or lithospheric thickness, and they adopted simplified disk load models of the Laurentian and Fennoscandian ice complexes. Ivins et al. [1993] extended this work by considering a set of global ice models, as well as regional subsets of each, and a suite of both layered and continuously (radially) varying viscosity profiles. Both studies assumed an incompressible rheology and uniform meltwater redistribution. [6] With the initiation of a new set of gravity missions, namely, CHAMP and GRACE, the accuracy of constraints on the spherical harmonic (zonal and nonzonal Stokes) coefficients of the secular change in the Earth s geopotential may be significantly improved [e.g., Dickey et al., 1997]. These impending constraints motivated the present study, in which we focus on a suite of sensitivity analyses related to the GIA signal. Recent GIA analyses have extended traditional numerical formalisms to include effects such as rotation-induced feedback [e.g., Milne and Mitrovica, 1996, 1998; Milne et al., 1999; Peltier, 1998, 1999]. We first explore the sensitivity of our predictions of the Stokes coefficients to several of these extensions. We next perform a more traditional sensitivity analysis in which we vary both the radial viscosity profile of the Earth model and the adopted late Pleistocene ice sheet history. This traditional analysis will involve both the individual Stokes coefficients and predictions of global maps of the secular geoid change. A companion paper extends the calculations performed here to consider Earth models that include lateral variations in mantle viscosity. [7] In practice, there are two possible applications of the predictions presented here. First, if all other signals associated with secular variations in the geopotential were known, then the predictions might be used to constrain rheology or ice history. Alternatively, one might use the predictions to correct observations of the secular changes in the geopotential for GIA and use the residual signal in some other geophysical analysis (e.g., constraining mass variations associated with ongoing hydrological, oceanographic and cryospheric processes). Our bias is toward the latter philosophy as long as the sensitivity analyses establish a plausible range for the correction, and thus a plausible uncertainty in the residual signal. 2. Secular Geoid Changes Due to GIA [8] In this paper we focus on perturbations to the geopotential and geoid (or sea surface) height. These perturbations are related via Brun s formula through a simple multiplicative factor equal to the surface gravitational acceleration [Lambeck, 1988]. Predictions of the geoid height anomaly due to GIA involve two distinct numerical steps. First, for a given viscoelastic Earth model and late Pleistocene ice history, the sea level equation is solved to determine a gravitationally self-consistent ocean load. In this study we adopt elements of the recent sea level theory outlined by Milne et al. [1999] for this purpose. In the second step, we convolve the total (ice plus ocean) surface load with the Green s function for the geoid height anomaly derived by Mitrovica and Peltier [1989]. This step also incorporates the influence of rotation perturbations on geoid height [Mitrovica et al., 2001]. All calculations are performed using spectral approaches with a truncation at spherical harmonic degree and order 64. [9] Let us denote the geoid height anomaly computed using the above procedure as G. In order to calculate the present-day (t = t p ) secular change of the geoid, _G, we use a finite difference method 1 _G q; f; t p ¼ G q; f; t p þ t G q; f; tp t ; ð1þ 2t as in the work by Mitrovica and Peltier [1993], where t was chosen to be 100 years and q and f are the colatitude and east longitude, respectively. _G can be expanded in terms of normalized Legendre functions ~P m [e.g., Chao and Gross, 1987] X _G 1 X q; f; t p ¼ a ¼0 m¼0 ~P m ðcos qþ : _C m t p cos mf þ _S m t p sin mf ; ð2þ where a is the radius of the Earth. The constants _C m and _S m are the Stokes coefficients, where is the degree and m is the order of the Legendre function. Past studies have concentrated on the values of the zonal harmonics, _J, which can be related to the values of _C 0 by p _J t p ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffi 2 þ 1 _C 0 t p : Also, _S 0 0. [10] Numerical predictions for the Stokes coefficients require the input of both an ice and Earth model. For the Earth model, we use elastic parameters given by PREM ð3þ

TAMISIEA ET AL.: SENSITIVITY OF SECULAR GEOID PREDICTIONS ETG 18-3 [Dziewonski and Anderson, 1981] and divide the mantle into two isoviscous layers, separated at the 670 km seismic discontinuity. The upper and lower mantle viscosities are denoted by n UM and n LM, respectively. In addition, the model includes an elastic lithosphere of thickness, h LT.In general, we will adopt a glacial cycle model based on the ICE-3G deglaciation history [Tushingham and Peltier, 1991] but modified to include a glaciation phase of duration 90 kyr. However, we also present predictions using a similarly modified version of the ICE1 deglaciation history [Peltier and Andrews, 1976]. Unless otherwise noted, two glacial cycles are used in all calculations. [11] In the sensitivity analyses described below it is impossible to present the complete set of Stokes coefficients for any reasonably broad degree range. (There are, for example, 49 such coefficients for spherical harmonic degrees up to 6.) As a result, in any specific figure we choose a subset (which will vary from figure to figure) that illustrates the main point of the calculations. 2.1. Effects of Common Simplifications [12] In Figure 1 we show predictions of various Stokes coefficients as a function of the lower mantle viscosity. The prediction of _C 20 has a form which is characteristic of numerous past predictions of the zonal harmonics when the upper mantle is fixed to 10 21 Pa s (see also the _C 31 case in Figure 1). At low viscosities the predicted signal is small because these models will have relaxed significantly, nearly reaching equilibrium, in the time (5 kyr) since the end of the model deglaciation. Models with high viscosity have progressively more sluggish relaxation throughout the postglaciation phase, and hence these models also yield progressively lower present-day signals. The predicted signals are thus highest at some intermediate value of the lower mantle viscosity, and the result is the profile shown in the top frame of Figure 1. For _C 20, the peak predicted amplitude is obtained for n LM 2 10 22 Pa s. [13] The results for the remaining, nonzonal harmonics in Figure 1 exhibit a variety of forms. The rather complicated sensitivity of the _C 21 and _S 21 predictions to variations in n LM is due to rotational effects. Changes in the rotation vector perturb the centrifugal potential, and this perturbation (known as the rotational potential) acts to load the planet. The rotational potential can have noticable effects on global geoid maps [Peltier, 1999; Mitrovica et al., 2001]. Because the rotational potential is dominated by true polar wander (TPW), it has a geometry governed by a spherical harmonic with = 2 and m = 1 [e.g., Han and Wahr, 1989]. Thus, one would expect the Stokes coefficients _C 21 and _S 21 to be perturbed the greatest due to this effect. This expectation is confirmed in Figure 1. Predictions of _C 21 and _S 21 without the rotational signal included (dashed lines) show the usual form associated with GIA results for nonrotating Earth models. The signal due to rotation is large enough to change the sign of the prediction relative to the nonrotating case. Furthermore, the sensitivity of the predictions to variation in n LM is similar to GIA-induced TPW speed predictions as a function of the same suite of Earth models [e.g., Yuen et al., 1986]. [14] Rotational effects are generally negligible for Stokes coefficients other than those at degree two and order one. Figure 1. Effect of rotational perturbations on predictions of various Stokes coefficients. Predictions are shown for a suite of lower mantle viscosities (n LM, on the abscissa scale), and in all cases the lithospheric thickness, h LT,is71 km and upper mantle viscosity, n UM,is5 10 20 Pa s. The elastic parameters are taken from PREM and the ice model includes two cycles of a modified version of ICE-3G (see text). The solid lines are predictions based on the theory for a rotating planet, while the dashed lines assume a nonrotating Earth (and thus ignore effects associate with perturbations in Earth rotation). The dotted line on the bottom frame is analogous to the solid with the exception that ocean loading effects are not included.

ETG 18-4 TAMISIEA ET AL.: SENSITIVITY OF SECULAR GEOID PREDICTIONS scale). Ocean loading effects contribute significantly in this case due to the geometry of the ocean basin. As a consequence, rotational perturbations to the ocean load also yield a nonnegligible contribution to the predicted signal. [15] A second simplification commonly employed in GIA studies is to assume that the Earth is elastically incompressible [e.g., Spada et al., 1992; Vermeersen and Sabadini, 1996]. This assumption has a particularly large effect on the _C 21 and _S 21 coefficients (Figure 2) and yields predictions Figure 2. The influence of assuming an elastically incompressible model on predictions of various Stokes coefficients. Predictions are shown for a suite of lower mantle viscosities (n LM, on the abscissa scale), and in all cases the lithospheric thickness, h LT, is 120 km and upper mantle viscosity, n UM,is10 21 Pa s. The elastic parameters are taken from PREM and the ice model includes one cycle of a modified version of ICE-3G (see text). The solid and dashed lines are predictions based on elastically compressible and incompressible Earth models, respectively. However, there are exceptions, and an example is given in the bottom frame of Figure 1. The dotted line on this frame shows a prediction for a rotating Earth in which all of the effects associated with the ocean load are removed. For _C 33 the total signal is very small (notice the ordinate Figure 3. Variation of selected Stokes coefficients with lithospheric thickness, h LT. Results are generated using the Earth models and ice history described in Figure 1 (n UM =5 10 20 Pa s), with the exception that we include the three cases: h LT = 71, 96 and 120 km (as labeled).

TAMISIEA ET AL.: SENSITIVITY OF SECULAR GEOID PREDICTIONS ETG 18-5 Figure 4. Sensitivity of _C m and _S m predictions up to degree and order 5 (as labeled) to variations in n UM and n LM. In all cases the lithospheric thickness, h LT, is 71 km. The elastic parameters are taken from PREM and the ice model includes two cycles of a modified version of ICE-3G. Each label at the top of the frame is proceeded by a minus sign if the prediction is negative and is followed by the multiplicative factor for the contour levels in parenthesis. The units are yr 1. The shaded overlay on each frame reflects the range of values for n UM and n LM inferred in recent analyses of GIA data (see text). that are over a factor of two too large at low values of n LM. Once again, this effect originates from the signal due to the rotational perturbation. Mitrovica and Milne [1998] have shown that TPW speeds for models with relatively weak lower mantle viscosity are significantly overestimated when an assumption of elastic incompressibility is introduced. This assumption affects the prediction of many of the other Stokes coefficients, though generally less than 10%. Indeed, the errors observed in the bottom three frames of Figure 2 are representative of the error introduced in many of the coefficients. 2.2. Earth and Ice Model Variations [16] Next, we examine the sensitivity of the predicted Stokes coefficients to variations in the Earth and ice model adopted in the GIA calculation. Previous studies have found that low degree zonal harmonics are generally insensitive to variations in lithospheric thickness [e.g., Mitrovica and

ETG 18-6 TAMISIEA ET AL.: SENSITIVITY OF SECULAR GEOID PREDICTIONS Figure 4. (continued) Peltier, 1993]. Figure 3 shows the prediction of selected Stokes coefficients as a function of n LM for three different values of h LT. Once again, _C 21 and _S 21 present a special case where the sensitivity evident on the figure originates from rotational effects. Predictions of GIA-induced TPW are sensitive to variations in the lithospheric thickness [e.g., Yuen et al., 1983; Peltier and Wu, 1983; Wu and Peltier, 1984; Mitrovica and Milne, 1998], particularly as the lower mantle viscosity of the Earth model is reduced [Mitrovica and Milne, 1998]. Specifically, thicker values of h LT increase the magnitude of the predicted TPW speed. For the other coefficients, the sensitivity to variations in h LT increases with an increase in lower mantle viscosity. As the degree of the Stokes coefficient increases, this trend becomes evident at lower values of n LM. These trends are straightforward to explain. As the degree of the coefficient or the lower mantle viscosity is increased, GIA-induced deformations become confined to progressively shallower depths. [17] In Figure 4 we provide contour plots showing predictions of Stokes coefficients for degrees two to five for a suite of Earth models in which both n UM and n LM are varied by nearly 2 orders of magnitude. For an upper mantle viscosity of 10 21 Pa s and for harmonics other than _C 21 and _S 21, the predicted signal peaks for an intermediate value of n LM and decreases relatively smoothly for smaller and greater values of the lower mantle viscosity. This simply confirms a general form evident in many prior studies. As the degree increases, the predicted peak amplitude is obtained for progressively lower values of n LM and the sensitivity to

TAMISIEA ET AL.: SENSITIVITY OF SECULAR GEOID PREDICTIONS ETG 18-7 n UM tends to increase. The sensitivity to upper mantle viscosity also increases as n LM is increased and n UM is weakened (and thus as flow is more confined to the shallow mantle). As an example, when n UM <2 10 20 Pa s, the sensitivity of many coefficients (e.g., _C 51 ) to variations in lower mantle viscosity disappears for n LM >10 22 Pa s, and in these cases the double root structure of the n LM sensitivity evident for the n UM =10 21 Pa s case also disappears. [18] Some consensus appears to be emerging in recent inferences of the average upper and lower mantle viscosity based upon GIA data [e.g., Mitrovica, 1996; Lambeck et al., 1998]. These studies suggest a value of n UM somewhat less than 10 21 Pa s, but greater than 10 20 Pa s, and n LM between 5 10 21 Pa s and 2 10 22 Pa s. We overlay, on Figure 4, a shading that reflects this range of current uncertainty. The relatively strong sensitivity to variations in n LM below 5 10 21 Pa s, evident for nearly all of the Stokes coefficients, is clearly avoided by limiting the viscosity model space in this manner. For most of the lowdegree harmonics this sensitivity is due to the sharp reduction in the present-day disequilibrium predicted for models in which n LM is reduced toward 10 21 Pa s. However, in the case of _C 21 and _S 21, the predictions increase as n LM is reduced below 5 10 21 Pa s as a consequence of a large increase in the predicted TPW signal. The highest amplitudes in Figure 4 are achieved for the zonal Stokes coefficients. However, we note that a set of the nonzonal harmonics reach relatively large values (e.g., _S 31, _S 41, etc.) and these may be strong candidates for detailed consideration using satellitederived constraints. [19] Uncertainties in the adopted late Pleistocene ice model also have potentially large effects on the predictions (for a detailed discussion, see Ivins et al. [1993]). In Figure 5 we show a set of predictions generated using both the ICE- 3G model and an ice model (ICE1B) modified from ICE1 [Peltier and Andrews, 1976] by the addition of an Antarctica loading cycle. The latter contribution was originally derived from Hughes et al. [1981] by Wu and Peltier [1983]. As we discussed in section 1, the large variation in the solution for _C 30 (and other odd zonal harmonics) is due to the sensitivity of this coefficient to the relative balance between the ice loads in the Northern Hemisphere and the Southern Hemisphere [Mitrovica and Peltier, 1993; Ivins et al., 1993]. Figure 5 indicates that predictions of the nonzonal harmonics are also sensitive to plausible variations in the ice model. In this regard, the differences apparent in Figure 5 are characteristic of variations we predicted for a much broader suite of Stokes coefficients. [20] In addition to providing a sensitivity analysis, Figures 3 5 also provide bounds on the GIA-induced contribution to low degree Stokes coefficients governing the secular variation of the geoid. These bounds will be useful for rigorous analysis of constraints emerging from the new set of satellite missions. These missions will constrain not only the Stokes coefficients but also regional variations of the geoid change, and so we turn next to predictions of the latter. [21] We computed global maps of the GIA-induced geoid change for a large suite of Earth models in which all combinations of the following values were adopted: h LT = 71, 96, or 120 km; n UM =1,2,3,5,8,or10 10 20 Pa s; and n LM = 2, 3, 5, 8, 10, 20, or 30 10 21 Pa s. In addition, Figure 5. Differences in various Stokes coefficients predicted using two cycles of ICE-3G and ICE1B (see text). Results are generated using the Earth models described in Figure 1 (n UM =510 20 Pa s, h LT = 71 km). for each of these Earth models, we computed results for both the ICE-3G and ICE1B ice histories. To proceed, we then culled this set of 252 predictions to include only those which satisfy a recent constraint on the present-day uplift rate in Hudson Bay (10.7 ± 1.4 mm yr 1 [Larson and van Dam, 2000]). This procedure yielded a set of 66 predictions, the simple mean of which is shown in Figure 6a. The geographic variation along 90 W longitude for the individual predictions is given in Figure 6b. The latter profile passes through the center of the Laurentide deglaciation,

ETG 18-8 TAMISIEA ET AL.: SENSITIVITY OF SECULAR GEOID PREDICTIONS Figure 6. Regional variations of secular geoid rates computed from a large suite of Earth and ice models. (a) Average secular geoid change, through degree and order 64, of predictions from 66 combinations of h LT, n UM, n LM, and ice model (see text). (b) Profiles of the predictions for each of the 66 models along 90 W longitude, as indicated by the arrows in Figure 6a. Predictions generated using ICE- 3G are shown in blue, while those using ICE1B are shown in green. The red line is the average value of the profiles, as shown in Figure 6a. and we adopt different colors to distinguish predictions based on the two ice histories. [22] The global map is characterized by regions of upwarping over the main centers of late Pleistocene deglaciation, including Laurentia, Antarctica and Fennoscandia. In the first two of these regions the average predicted signal exceeds 1 mm yr 1. In addition, there is a distinct Y 21 pattern associated with rotational (TPW) effects that is evident in the far-field geoid signal. In the absence of the TPW signal there would be a broad subsidence of the geoid in the far-field. The rotation signal adds to this subsidence in the vicinity of Central America and southern Australia and the net effect is an average peak subsidence that exceeds 0.5 mm yr 1. Clearly, the GIA-induced geoid rate

TAMISIEA ET AL.: SENSITIVITY OF SECULAR GEOID PREDICTIONS ETG 18-9 is nonnegligible even in the far field of the late Pleistocene ice cover [Mitrovica and Peltier, 1991; Peltier, 1999; Mitrovica et al., 2001]. [23] Predictions of the geoid rate over Laurentia and Antarctica are clearly sensitive to variations in the adopted ice and Earth models (Figure 6b). However, this sensitivity remains significant in the far field, where predictions have a range of 0.5 mm yr 1 (or 0.25 mm yr 1 around the average). When regional maps of geoid rates become available, efforts will be made to correct these maps using some standard model for GIA. The results in Figure 6b indicate that this correction may be subject to a relatively large uncertainty, even within the far-field. 3. Summary [24] Previous studies have shown that the predicted zonal harmonics of the secular geoid change due to GIA are sensitive to variations in the adopted Earth and ice model. We have extended this conclusion to nonzonal harmonics and have provided a detailed sensitivity analysis involving variations in both the main input parameters (mantle viscosity, lithospheric thickness, ice history) and elements of the forward theory (Earth rotation, elastic compressibility). Some Stokes coefficients show sensitivities akin to those identified in earlier zonal analyses, while others show distinct sensitivities originating, for example, from rotational effects (e.g., _C 21 and _S 21 ) and ocean loading (e.g., _C 33 ). Regardless of the source of the sensitivities, our analysis provides bounds on the variation of these coefficients which will be important as new values are derived from upcoming satellite missions. [25] The same reasoning motivated our detailed analysis of regional predictions of the secular geoid change. We find that these predictions are also highly sensitive to variations in the Earth and ice model. Indeed, even in the far field of the late Pleistocene ice sheets this variation is 0.5 mm yr 1 or greater. Because long time series will be required to constrain GIA from (for example) GRACE data, it will be difficult to robustly infer a GIA Earth/ice model combination from the data before the end of the mission. Thus it is more likely, at least in the short term, that the analysis of this data will proceed by correcting the data using some independently derived standard model. In this case, the ranges we have established may serve as a measure of the uncertainty in this correction. [26] There are at least two areas of this research that warrant further study. The first of these concerns predictions that involve a nonnegligible contribution from TPW effects (in particular, the coefficients _C 21 and _S 21, and the Y 21 imprinting evident on maps such as Figure 6a). Recent GIA analyses indicate that predictions of polar wander driven by surface mass loading are sensitive to the treatment of the density discontinuity at 670 km depth (nonadiabatic, as in this study, versus adiabatic (e.g., Mitrovica and Milne [1998] and, especially, Johnston and Lambeck [1999, Figure B1]) and the adoption of realistic viscoelastic stratification within the lithosphere [Nakada, 2000]. In regard to the latter, the validity (and accuracy) of our assumption of a global elastic lithosphere for TPW predictions on GIA timescales is uncertain. [27] The second concern involves the adoption of spherically symmetric Earth models. Upper mantle (including lithospheric) asymmetry between ocean and continent environments, as well as thermochemical variations driven by mantle convection, imply significant lateral variations in mantle viscosity. In a future paper we will examine the sensitivity of the predictions to these lateral variations. 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(g.a.milne@ durham.ac.uk) J. X. Mitrovica and M. E. Tamisiea, Department of Physics, University of Toronto, 60 St. George St., Toronto, Ontario, Canada M5S 1A7. ( jxm@physics.utoronto.ca; tamisiea@physics.utoronto.ca) J. Tromp, California Institute of Technology, Seismological Laboratory, Mail Code 252-21, 1200 E. California Blvd., Pasadena, CA 91125, USA. ( jtromp@gps.caltech.edu)