Glacial isostatic adjustment and relative sea-level changes: the role of lithospheric and upper mantle heterogeneities in a 3-D spherical Earth

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1 Geophys. J. Int. (26) 165, doi: /j X x Glacial isostatic adjustment and relative sea-level changes: the role of lithospheric and upper mantle heterogeneities in a 3-D spherical Earth G. Spada, 1 A. Antonioli, 2, S. Cianetti 2 and C. Giunchi 2 1 Istituto di Fisica, Università di Urbino Carlo Bo, Via S. Chiara 27, Urbino, Italy. spada@fis.uniurb.it 2 Istituto Nazionale di Geofisica e Vulcanologia, Via di Vigna Murata 65, I Roma, Italy Accepted 26 February 23. Received 26 February 23; in original form 25 May 26 GJI Tectonics and geodynamics SUMMARY The response of the Earth to the melting of the Late Pleistocene ice sheets is commonly studied by spherically layered models, based on well-established analytical methods. In parallel, a few models have been recently proposed to circumvent the limitations imposed by spherical symmetry, and to reproduce the actual structure of the lithosphere and of the upper mantle. Their main outcome is that laterally varying rheological structures may significantly affect various geophysical quantities related to glacial isostatic adjustment (GIA), and particularly post-glacial relative sea-level (RSL) variations and 3-D crustal velocities in formerly ice-covered regions. In this paper, we contribute to the ongoing debate about the role of lithospheric and mantle heterogeneities by new 3-D spherical Newtonian finite elements models and we directly compare their outcomes with publicly available global RSL data. This differs from previous investigations, in that have mainly focused on extensive sensitivity analyses or have considered a limited number of RSL observations from formerly glaciated regions and their periphery. In our study the lithospheric thickness mimics the global structure of the cratons based on geological evidence, and the upper mantle includes a low-viscosity zone beneath the oceanic lithosphere. We use two distinct global surface loads, based upon the ICE1 and ICE3G deglaciation chronologies, respectively. Our main finding is that using all of the available RSL observations in the last 6 years it is not possible to discern between homogeneous and heterogeneous GIA models. This result, which holds for both ICE1 and ICE3G, suggests that the cumulative effects of laterally varying structures on the synthetic RSL curves cancel out globally, yielding signals that do not significantly differ from those based on the 1-D models. We have also considered specific subsets of the global RSL database, sharing similar geographical settings and distances from the main centres of deglaciation. When we consider the data from the margins of the Baltic region, a laterally varying lithospheric thickness improves significantly the agreement with the observations. This is not observed in other relevant situations, including the Hudson bay region. In the regions where the disagreement between predictions and observations is particularly evident, further investigations are needed to improve the geometry of the heterogeneous structures and of the surface ice-sheets distribution. Key words: glacial rebound, mantle viscosity, sea-level variations. 1 INTRODUCTION The importance of lithospheric and mantle heterogeneities in glacial isostatic adjustment (GIA) problems has been the subject of extensive investigations for many years. The development of new numerical techniques, the increase in computational resources and the availability of high-quality relative sea-level (RSL) and geodetic Now at: Geophysics Research Group, School of Environmental Science, University of Ulster, Coleraine, BTS2 1SA, UK. data provide strong motivations to continuously improve GIA models. The first attempts at modelling post-glacial rebound in the presence of heterogeneities were carried out either in plane strain or in axis-symmetric approximations (see e.g. Gasperini & Sabadini 1989; Giunchi et al. 1997; Kaufmann et al. 1997; Giunchi & Spada 2; Wu 22a). However, the GIA, due to the complex ice load evolution and the lateral variations in the rheological properties of the Earth, is a fully 3-D, global problem. The findings by Kaufmann & Wolf (1999) and Wu (22a) on mode coupling in models with laterally varying rheological properties suggested the development of perturbative methods, valid for small 692 C 26 The Authors Journal compilation C 26 RAS

2 GIA and RSL in a spherical 3-D earth 693 rheological contrasts (Tromp & Mitrovica 2), and the application of various numerical techniques (Martinec 1999; Zhong et al. 23; Wu 24; Latychev et al. 25a) aimed to tackle the GIA problem in three dimensions. A great deal of literature is devoted to the study of post-glacial rebound in a flat Earth using finite element (FE) approaches in two or in three dimensions. Some studies (see e.g. Dal Forno et al. 25) focused on mantle rheology, evaluating the performance of linear versus non-linear, dislocation creep rheologies in the upper mantle. In other investigations, the FE method has been employed to model lateral variations of lithospheric thickness and asthenospheric viscosity in a fully Newtonian earth. Among these, Kaufmann et al. (2), Kaufmann & Wu (22) introduced, for the Fennoscandian ice sheet, a realistic distribution of ice and mantle heterogeneities. The above studies have confirmed that predictions of past and present glacial signatures are significantly influenced by lateral variations in lithospheric thickness, and particularly by viscosity heterogeneities in the asthenosphere. The introduction of spherical 3-D GIA models is necessary to account for realistic lateral variations in the rheology of the mantle and to reproduce correctly the post-glacial sea-level variations. The most relevant contributions (Wu 22b; Zhong et al. 23; Wu 24; Latychev et al. 25b,a), based on FE or finite-volumes methods, are mainly concerned with lateral variations of the lithospheric thickness, whose geometry has been prescribed according to various criteria. For example, Zhong et al. (23) investigated the effects of a laterally varying oceanic lithosphere defined by the 75 C isotherms, taking into account the thinning in proximity to the spreading centres and the thickening in the basins. A similar approach was followed by Latychev et al. (25b), who considered the effect on GIA observations of the rheological discontinuity along the mid-ocean ridges in a spherical Earth. The objective of our work is twofold. First, using the FE approaches mentioned above, we introduce a new simplified global model that accounts simultaneously for the lithospheric and upper mantle heterogeneities suggested by the global model 3SMAC (Nataf & Ricard 1996). Second, for the first time, we assess their impact on the interpretation of the post-glacial Holocene RSL variations reported by a global dataset; this is done by a quantitative misfit analysis either at a regional and at a global scale. The paper is organized as follows. The proposed models and the details of the numerical approach are described in Section 2. In Section 3 we show the predicted RSL at selected sites located in different glaciological regions, whereas a discussion of the significance of the results is presented in the final section of the manuscript. 2 METHODS, NUMERICAL APPROACH AND VALIDATION In this work we simulate post-glacial rebound by means of a 3-D FE approach solving the equilibrium equations of an incompressible, non-self-gravitating, spherical Earth. Based on previous experience (Giunchi & Spada 2), we have used the commercial package MARC throughout (MSC.Marc 23). Our approach here is similar to that by Wu & van der Wal (23) since we employ a laterally heterogeneous spherical model, but we do not solve for the gravitationally self-consistent sea-level equation () (Farrell & Clark 1976) in its complete form (the errors implied in neglecting the selfconsistent ocean load are discussed in Wahr & Davis (22) and Wu & van der Wal (23), and will be also addressed starting from Section 3 below). While Wu et al. (25) has limited his attention to the effects of a laterally varying lithosphere upon relative sea-level change, we also account for 3-D rheological heterogeneities related to the ocean-continents distribution. Latychev et al. (25a) have recently considered, for the first time, a global-scale low-viscosity narrow region (such as mid-ocean ridges) in the framework of 3-D spherical models of GIA. In our model the Earth is discretized into eight-node isoparametric brick elements whose shape is determined by the cubed sphere mapping of Ronchi et al. (1996). The computational domain extends from the surface of the Earth down to the core mantle boundary, and is characterized by a horizontally non-uniform resolution to optimize the CPU time. The mesh has a resolution of 2 2 underneath the Laurentian and Fennoscandian ice sheets, where the post-glacial deformations are expected to be the largest. A coarser grid is used outside these regions, with a resolution of 5 5. Appropriate contact elements ensure the continuity of the displacement field at the boundaries between the fine and the coarse portions of the mesh. In the radial direction, the resolution is maximum at the surface ( 1 km) and decreases with increasing depth ( 5 km). The vertical and horizontal resolutions employed allow us to satisfactorily reproduce the deformations predicted by laterally homogeneous models, as shown in Fig. 1 below. All the FE models developed share the same boundary conditions. To simulate the isostatic restoring forces, Winkler foundations are applied at all the material interfaces characterized by a density contrasts (Williams & Richardson 1991; Wu 24). Winkler force is computed as g i δρ i, where g i and δρ i are the gravitational acceleration and the density contrast across the ith interface, respectively. This formulation is widely employed in FE approaches to GIA (see e.g. Wu 24). Despite the models proposed in this paper may include a laterally heterogeneous viscosity structure, they are always characterized by a radially stratified density to fulfil the Winkler formulation. The density and rigidity profiles used in the present paper are reported in Table 1. Fig. 2 shows the four models of the lithosphere and shallow upper mantle considered in this study. All of them are characterized by a radially stratified structure beneath the 42-km-depth discontinuity, a Newtonian viscoelastic rheology for the mantle, an elastic lithosphere, and an inviscid uniform core. The first model (), characterized by a uniform lithosphere and a uniform mantle, is the FE implementation of the semi-analytical five-layer, 1-D homogeneous model formerly used by Cianetti et al. (22), even though here the thickness of the elastic lithosphere is fixed to 1 km. includes three mantle layers, labelled by SM (shallow upper mantle), TZ (upper mantle), and LM (lower mantle), respectively. The viscosity profile of the mantle, also reported in Table 1, is that implied in the construction of the ice chronology model ICE3G (Tushingham & Peltier 1991). In the concluding section we will also consider the case of a more viscous lower mantle, with viscosity of 1 22 Pa s. The second model considered in this study () is characterized by a laterally variable lithospheric thickness (Fig. 2) adopted from the model 3SMAC (Nataf & Ricard 1996), that reproduces the large-scale structure of continents and Archean cratons. Accordingly, we assume a 1-km-thick oceanic lithosphere, a 2- km-thick continental lithosphere and 3-km-thick cratonic regions see Fig. 3). All of these lithospheric regions share the same physical parameters and are only characterized by a different thickness. In previous investigations, different lithospheric structures have been assumed, inferred from recent shear wave tomographic models (Wu et al. 25), by the 75 C geotherm (Zhong et al. 23), or by the global network of plate boundaries (Latychev et al. 25a). In the third model (), the lithosphere is uniform whereas the mantle is characterized by the laterally variable viscosity introduced C 26 The Authors, GJI, 165, Journal compilation C 26 RAS

3 694 G. Spada et al Richmond 18 Ungawa Pen. TABOO FE 233 Angermanland 235 Helsinki Table 1. Geometrical and rheological parameters. a Region Radius Density Rigidity Viscosity Gravity km kg m 3 GPa 1 21 Pa s m s 2 LITHO SM OM TZ LM CORE a With SM, OM, TZ, and LM we indicate the shallow upper mantle, the oceanic mantle, the transition zone, and the lower mantle, respectively. A sketch of these regions is given in Fig Tay Sound 323 Boston 22 Varanger Fjord 29 Bjugn lithosphere: model is combined with the oceanic structure of. Although in our ensuing investigations we will mainly gauge the effects of lateral heterogeneities on GIA, we will also address how uncertainties in the chronology of the surface load affects the RSL curves in the presence of 3-D structures. For this reason, we have considered two different ice chronologies: ICE1, based upon glaciological evidence (Peltier & Andrews 1976), and ICE3G, a considerably more refined chronology built to maximize the agreement with the RSL global dataset assuming a specific mantle viscosity profile (Tushingham & Peltier 1991). Both models have been modified to include a 1--long ice accumulation stage during which the load builds at a constant rate until it reaches the last glacial maximum (LGM) 18 BP. The contours of the two ice aggregates at the LGM are shown in Fig. 3. In the FE analysis, we compute the sea-level change S according to Malden Is New Caledonia -6-3 Figure 1. Analythical (solid) (Spada et al. 24) versus FE (dotted) RSL curves for a three-layer test model. The model includes a 1-km-thick elastic lithosphere (ρ = 412 kg m 3, μ = 73 GPa), a uniform viscoelastic mantle (ρ = 458 kg m 3, μ = 2 GPa, η = Pa s), and inviscid core with radius of 348 km (ρ = 1925 kg m 3 ). by Cadek & Fleitout (23) in the framework of global geodynamics. In agreement with Cadek & Fleitout (23), in model the viscosity ratio SM/OM is 1 2. Such a thick low-viscosity region is expected to provide an upper bound for the effects of the rheology of the sub oceanic mantle on relative sea level. As for, the characteristic function of the continents is adapted from 3SMAC. The fourth model considered here () contains both lateral viscosity variations in the shallow upper mantle and an inhomogeneous S(θ,λ,t) = U(θ,λ,t), (1) where U is the vertical displacement of the solid surface of the Earth, θ and λ denote the colatitude and longitude of a given site, and t is time. Eq. (1) can be obtained from the self-consistent assuming (i) a steady state ice mass, (ii) a vanishing water density and (iii) a non-self-gravitating solid Earth (e.g. Spada & Stocchi 26). Therefore, in our ensuing computations, we will limit our attention to the process of free relaxation of the Earth s surface that followed the end of ice melting and we will neglect the ocean load. Focusing to the post-glacial phase minimizes the errors expected by neglecting self-gravitation, whose implementation is currently under way. The validity of eq. (1) has been discussed by Wahr & Davis (22) and by Wu & van der Wal (23) and will be further addressed in Section 3 by comparisons with computations that were obtained by solving the full version of the for a spherically symmetric Earth. According to the eq. (1), the synthetic RSL curves are constructed as: RSL(θ,λ,t BP ) = [U(θ,λ,t BP ) U(θ,λ,t P )], (2) where t BP is time before present and t P is present time. In Fig. 1 we benchmark the RSL curves obtained by the FE model using ICE1 with those computed by a semi-analytical method (Spada et al. 24), based on spectral decomposition of the vertical displacement truncated at the harmonic degree l max = 72. For this computation, we have employed a three-layer test model, whose properties are given in the figure caption. FE-based and analytical predictions are shown by dotted and solid curves, respectively. The agreement between the two methods is satisfactory in all of the RSL sites considered, but a significant discrepancy is observed in the case of Tay Sound (NW Territories) and Bjugn probably due to an C 26 The Authors, GJI, 165, Journal compilation C 26 RAS

4 GIA and RSL in a spherical 3-D earth 695 ocean oce an ocean contine nt ocean crato n contine nt crato n oce an oce an LITHO OM SM TZ Figure 2. Homogeneous () and heterogeneous models (,, ) considered in the paper. The lower mantle region, not depicted here, has the same properties in all of the four models shown. The physical and rheological properties of the various regions are given in Table been chosen according to their location with respect to the former ice sheets. They include Figure 3. Lithospheric thickness according to model 3SMAC (Nataf & Ricard 1996), implemented in our GIA heterogeneous models. The thickness of the oceanic (light grey), continental (medium grey) and cratonic lithosphere (dark grey) is assumed to be 1, 2 and 3 km, respectively. The top and bottom frames also show, by white solid lines, the contours the ICE1 and ICE3G ice chronologies at the last glacial maximum, respectively. inaccurate discretization of the ice model in the proximity of the ice margins. 3 P R E D I C T E D R E L AT I V E S E A L E V E L In Figs 4 8 we show predictions at selected sites belonging to areas characterized by broadly similar post-glacial RSL trends that have C 26 The Authors, GJI, 165, C 26 RAS Journal compilation (i) regions close to the centre of the loads (the Hudson Bay and the Baltic regions), (ii) regions located at the periphery of the ice sheets (Northwestern Territories, the Eastern coast of the United States, and the coasts of Fennoscandia) and (iii) the extreme far-field region of the Pacific Islands. As shown in Fig. 3, these areas are also characterized by different positions with respect to the heterogeneous structures introduced in our modelling. The centres and the peripheries of the previously glaciated regions fall close to the central portions of the Archean cratons and to their borders, respectively, whereas the Pacific area is entirely comprised in the oceanic lithosphere domain. The results obtained for region (i) will be presented in Section 3.1 below, with regions (ii) and (iii) discussed in Section 3.2. To deal with a uniform database, all of the data and associated errors are taken from the compilation of Tushingham & Peltier (1993), hereinafter TP, despite the fact that we are aware that some of these data, such as those in the Hudson Bay region, were later subject to a significant revision (Mitrovica & Peltier 1995). Among the available RSL sites in any given region, we chose to show predictions for those that collect more than one datum in the last 6, and whose trend can be considered representative of the whole region. As discussed in Section 2 above, our analysis is restricted to this time period due to the limitations of our FE approach. To compare the sensitivity of the RSL curves to both the laterally varying rheology and the surface ice load, we will, in the following, consider both the ICE1 and the ICE3G chronologies. This local study will then be extended to a regional and global scale in Section Ice centre Here we discuss the role of lateral heterogeneities on the RSL curves pertaining to the Hudson Bay and Baltic regions, as these are the most representative and well studied among the formerly ice-covered regions. Fig. 4 shows predictions and observations for two of the eight sites belonging to Hudson Bay (Richmond Gulf in Figs 4a and c and Ungawa peninsula in Figs 4b and d), marked

5 696 G. Spada et al. 11 Richmond Gulf 18 Ungawa Pen ICE3G RSL (m) Figure 4. RSL predictions from,,,, and models in the Hudson Bay area sites: results for Richmond Gulf and Ungawa Peninsula (see left map) are shown in panels, for the ICE1 and, for the ICE3G models, respectively. Crosses show data and associated 1σ errors from the Tushingham & Peltier (1993) dataset. 233 Angermanland 235 Helsinki ICE3G RSL (m) Figure 5. RSL predictions for the central portion of the Baltic region. The locations of Angermanland and Helsinki are shown in the left map. by stars in the map. The top and bottom frames pertain to the ICE1 and ICE3G chronologies, respectively. The four RSL curves correspond to each of the four models described in Section 2; with solid black lines always used to show results based upon the spherically symmetric model (). With light grey curves labelled by we show synthetic RSL curves obtained solving the gravitationally self-consistent assuming a radially stratified earth model as in. The has been solved by the pseudospectral method of Mitrovica & Peltier (1991). Due to the regional uplift caused by the melting of the Laurentian ice sheet, the data shown in Fig. 4 indicate a clear sea-level fall in the last 6. The various synthetic RSL curves fall close to each other, their offsets being comparable with the data uncertainties. We have verified that this also holds for the remaining six sites of the Hudson Bay, as discussed in the regional analysis of Section 4. This indicates that the Hudson Bay RSL data cannot constrain the extent and geometrical features of heterogeneous models, and supports previous studies where different laterally varying structures are implemented (Zhong et al. 23; Latychev et al. 25a) and the sea-level variations are computed in a gravitationally self-consistent manner (Wu et al. 25). In particular, Fig. 4 shows that, independently from the ice model employed, predictions based upon (dashed) cannot be distinguished from those obtained by (solid). This lack of sensitivity to the rheological contrast between the continental and C 26 The Authors, GJI, 165, Journal compilation C 26 RAS

6 GIA and RSL in a spherical 3-D earth Tay Sound 323 Boston ICE3G RSL (m) Figure 6. The same as in Fig. 4, but for NW territories and the Atlantic coast of the United States, respectively. Predictions are shown for the sites of Tay Sound and Boston. 22 Varanger Fjord 29 Bjugn ICE3G RSL (m) Figure 7. Predictions of RSL for two sites of the margins of the Baltic region, Varanger Fjord and Bjgun, respectively. the oceanic regions is a consequence of the relatively large distance of the Hudson Bay RSL sites from the heterogeneities imposed in this study (see Fig. 3). This will be further discussed in Section 4. It is also apparent that the RSL data in the two sites cannot be simultaneously matched by the spherically layered model or by any of the heterogeneous ones. This is an indication that the known inconsistency of the Hudson Bay RSL data (Mitrovica & Peltier 1995; Cianetti et al. 22) cannot be adequately resolved by the long wavelength heterogeneities considered here. Comparing the top with the bottom panels, we can observe the greater sensitivity of RSL predictions to the ice history model compared to lithospheric thickness and rheological lateral variations. As a further point, from frames and of Fig. 4 we note that the ICE1 predictions for models with lateral variations in the lithospheric thickness (VL) depart, albeit slightly, from those characterized by a homogeneous lid (UL). On the other hand, when ICE3G is employed (Figs 4c and d), the four RSL curves fall very close to each other. This result can be attributed to the different surface mass distributions of the Laurentide ice sheet in ICE1 and ICE3G (Peltier & Andrews 1976; Tushingham & Peltier 1991). While the two aggregates have comparable total mass at the LGM (close to kg), in ICE3G the Laurentian reaches its maximum thickness of 32 m in a narrow region at the southernmost point of the Hudson Bay, while ICE1 is characterized by a larger maximum C 26 The Authors, GJI, 165, Journal compilation C 26 RAS

7 698 G. Spada et al. 679 Malden Is. 671 New Caledonia ICE3G RSL (m) Figure 8. Expected and observed RSL variations for two representative sites of the Pacific Islands (Malden Island and New Caledonia). thickness ( 36 m) and maintains a thickness in excess of 32 m on a broad region along the whole southwestern portion of the Bay. For these reasons, ICE1 can probe a deeper and larger portion of the mantle, so that the pattern of the resulting internal flows are more influenced by the heterogeneous structure of the lithosphere. We also notice that predictions obtained using the (grey), based on the assumption of a laterally homogeneous model, provide the best match with the RSL observations for the two sites considered here. However, it is also apparent that uncertainties about the surface load largely exceed the error made neglecting hydro-isostasy. These points, as well as the performances of with respect to, will be addressed in Section 4 both for the whole set of data available for this region, as well as on global scale. The RSL predictions for the central portion of Fennoscandia are shown for the two representative sites of Angermanland (Figs 5a and c) and Helsinki (Figs 5b and d). We notice that, independently from the ice load and heterogeneous structure adopted, the RSL curves for the UL models virtually coincide, similar to what we have already found for the Hudson Bay region (see Fig. 4). This lack of sensitivity to lateral variations of mantle viscosity can be attributed to the small size of the Fennoscandian ice sheet compared to the depth where these variations are assumed to occur in this study and to their lateral extent (see Fig. 3). In addition, curves UL significantly exceed those of the kind VL in the whole time span considered. This is a consequence of the fact that a thicker lithosphere produces less deformation within the ice margin because it spreads the deformation farther away from the load. The minor role of the shallow upper mantle lateral viscosity variations in this central region is also confirmed when a lithosphere of variable thickness is adopted (compare curves with ). The results of Fig. 5 can be used to discuss the outcomes of previous investigations based on axis-symmetric craton models about the moderate sensitivity of the uplift data to the presence of anomalies in the lithospheric thickness for sites close to the rebound centre in Fennoscandia (see e.g. Gasperini & Sabadini 1989; Kaufmann et al. 1997, and references therein). Our 3-D computations allow us to better appreciate how this sensitivity varies according to the distance from the centre of the rebound and the trade-off between the ice-sheets chronology and the mantle heterogeneous structures. In the specific case of Helsinki (Figs 5b and d), the offset between the UL and VL curves largely exceeds the data uncertainties if ICE1 is employed eq. (5b), whereas for Angermanland this is observed for both the ice loads (frames a and c). This is in qualitative agreement with the results by Zhong et al. (23), who observed that the central sites of Fennoscandia are more sensitive to the anomalous lithospheric thickness compared to central North America. From Fig. 5 we observe that the introduction of the 3-D model hinders the detection of systematic trends in the synthetic RSL curves, since each single uplift time history has a unique relation with the geometry of the problem (i.e. the ice load details and the structure of the heterogeneities) and its rheological aspects as well (i.e. the viscosity values in the anomalous regions). A better insight into the effects of 3-D rheology upon the RSL sites belonging to a specific region can be gained in the analysis of Section 4. As a final remark, we notice that with the exception of Angermanland (frame a) predictions based on the (grey shaded curves) fall within 1 σ from the curves, indicating that in this central region eq. (1) is valid, at least in the framework of spherically layered earth models. As we will discuss in Section 4 below, the curves are those that globally better fit the data in this region when ICE3G is employed, as it is clearly shown here in the case of Angermanland. 3.2 Periphery and far field As examples of sites belonging to the margins of the former Laurentian ice sheet, we have selected Tay Sound (Baffin Island, Northwestern Territories) and Boston (North American Atlantic coast). The results are shown in Fig. 6, where the map shows the location of the RSL sites. Due to the irregular shape of the ice margins and of the mantle heterogeneities beneath these regions (see Fig. 3), the detection of trends from a limited number of RSL curves is even more difficult than in the case of the central RSL sites considered in Section 3.1 as it is evidenced also by the poor performance of the benchmark model in Tay Sound, Fig. 1. For the peripheral C 26 The Authors, GJI, 165, Journal compilation C 26 RAS

8 GIA and RSL in a spherical 3-D earth 699 sites considered here, both the predictions and the sensitivities to the rheological structures vary considerably according to the ice chronology. When ICE1 is employed, the Tay Sound RSL observations (Fig. 6a) are better reproduced by the models of the type VL, while none of the FE predictions based upon ICE3G (eq. 6c) can explain the data. The observations pertaining to Boston (eq. 6b) indicate a monotonous sea-level rise in the last 6, which can be attributed to the collapse of the peripheral bulge surrounding the formerly glaciated regions. Similar to Tay Sound, Boston is close to the transition between the Archean craton and oceanic lithosphere, a region of considerable complexity according to the structural model adopted here (see Fig. 3). When ICE1 is adopted (eq. 6b), only model can reproduce the observed monotonous sea-level rise. However, using ICE3G (eq. 6c) all of the predictions (including ) show the correct trend, and the data are better reproduced by models including lateral variations of the lithospheric thickness (VL models). The sensitivity to the ice chronology that characterizes the two sites considered here is due to both the distinct surface mass distributions of ICE1 and ICE3G at the LGM, and to the details of their melting chronology close to their margins. As we have verified, this sensitivity is shared by most of the sites of the Northwestern Territories and the North American Atlantic coast, where the two ice models predict significantly distinct time evolutions. The trade-off between ice chronology and mantle heterogeneities is also evident in Fig. 7 that portrays predictions for two representative sites at the margins of the former Fennoscandian ice sheet, namely the Varanger Fjord and Bjugn (see map). Both sites are relatively close to the transition between the Archean and the oceanic lithosphere that, according to our model of Fig. 3, occurs along the Atlantic margins of Norway. For this reason we have observed that, for most of the sites belonging to this region, deviations from the predictions of the laterally homogeneous model are quite significant. This is well documented by the two examples of Fig. 7, suggesting an important role of lateral variations of the lithospheric thickness, as previously reported in a number of studies (see e.g. Kaufmann et al. 1997). From the sole analysis of these two sites, however, it is not possible to determine whether these peripheral sites are more or less sensitive than the central ones to the 3-D structure of the mantle beneath the load. We observe that even though the sealevel fall characterizing these sites is well reproduced by all of the four FE models employed, the predictions are considerably scattered with respect to the available data. When ICE3G is adopted, the (that implies a laterally homogeneous mantle) produces the better agreement with the observations, whereas in the case of ICE1 (top frames) these are poorly fitted. As in the case of the NW Territories (see Fig. 6 above), the predictions are found to be significantly sensitive to the ice chronology. This is a common feature to all of the sites of this region, as it results from the regional analysis of Section 4. Fig. 8 shows the RSL curves for two far-field sites located in the equatorial Pacific ocean (Malden Island and New Caledonia, respectively). The RSL data pertaining to these sites indicate a sealevel highstand of a few metres in the last 6. This is common to all of the sites in this portion of the Pacific ocean, corresponding to zone V defined by Clark et al. (1978). As discussed by Mitrovica & Peltier (1991) and Mitrovica & Milne (22), this feature of the RSL curve can be attributed to the phenomenon of the equatorial ocean siphoning, in which water masses migrate from the ocean far-field regions into the near-field regions to compensate for the collapse of the peripheral bulges surrounding the formerly glaciated areas. This process is qualitatively well reproduced by the predictions, based upon the solution of the (grey solid curves), which account for the observed sea-level fall in the Late Holocene. However, due to the lack of a correct description of the ocean load, our FE solutions [that are based on the approximation (1)] mostly predict an opposite trend, characterized by a slow monotonous post-glacial sea-level rise. This holds for all of the sites of this region. The failure of the numerical approach in the far-field regions prevents robust conclusions to be drawn from the results shown in Fig. 8, but we nevertheless observe that in spite of their large distance from the continental margins, the synthetic RSL curves show sensitivity to the model employed, with distinct predictions for models UM and VM. Although the sensitivity to the lateral structure of the mantle does not exceed the data uncertainty, this is a first indication of the importance of the mantle lateral structure upon the actual RSL observations in oceanic areas that merits further investigations. 4 DISCUSSION AND CONCLUSIONS To summarize the results obtained, we perform a region-by-region study aimed to compare the performances of the five models employed in the previous section. In addition to the areas considered in Sections 3.1 and 3.2 above, in Fig. 9 we also show the regional misfits for the Mediterranean and the British Islands. A global analysis is performed in Fig. 1, where the TP database is considered as a whole. For a given ice-sheet chronology, we compute the misfit for the ith site according to: M i = 1 N i ( d c k d o ) 2 k, (3) N i σ k=1 k where N i is the number of data available for the site, dk o, σ k and d c k are the observed datum, its standard deviation and the corresponding prediction, respectively (Cianetti et al. 22). A regional misfit is subsequently obtained by averaging the M i values as: M r = 1 N s M i, (4) N s i=1 where N s is the number of sites considered. Each frame of Fig. 9 shows the ranges of labels of the sites belonging to each region, as they are given in the TP database. To be consistent with the analysis of Section 3, in eq. (4) we have only considered the sites with at least one datum in the last 6, and to avoid biases produced by outliers we do not include the contributions from the sites characterized by the smallest and the largest misfits. For these reasons, N s is always smaller than the number of sites available in the database for that region. The value of N s is indicated in parentheses at the top of each frame. Since we are not performing an exhaustive exploration of the parameters space (the geometry and the viscosity of the various rheological regions are kept fixed to reference values), the analysis that follows has the sole purpose of evaluating the relative performances of the proposed models; a rigorous inversion of the 3-D heterogeneities from RSL data is beyond the scope of this paper. As undertaken by Cianetti et al. (22) and Piana Agostinetti et al. (24) in a similar context, a simple F-test will be applied to evaluate the statistical significance of the results. From the results of Figs 9 and 1 we can draw four main conclusions that concern (i) the performance of ICE3G compared to ICE1, (ii) the relevance of lateral variations of the lithospheric thickness and of lateral viscosity variations, (iii) the importance of employing the for RSL predictions and (iv) the role of mantle heterogeneities on a global scale. C 26 The Authors, GJI, 165, Journal compilation C 26 RAS

9 7 G. Spada et al. (i) A clear feature of Fig. 9 is the better performance of ICE3G (crosses) compared to ICE1 (circles) when model is employed. This is particularly evident when we consider the data from the Hudson Bay, NW Territories, Atlantic coast, and Baltic (margins) regions, where the misfit reduction observed when ICE1 is substituted with ICE3G is significant at the 99 per cent confidence level. In general, with increasing distance from the formerly glaciated areas, the differences between the regional misfits of ICE1 and ICE3G become less pronounced. The overall better performance of ICE3G is not surprising, since this chronology was indeed built to improve the agreement of the previous models ICE1 and ICE2 with the RSL observations worldwide assuming a radially stratified model (Tushingham & Peltier 1991). The new finding here is that ICE3G is generally superior to ICE1 even when laterally varying structures are accounted for, and the is solved according to our approximation (eq. 1). Furthermore, the predictions based upon ICE3G are generally found to be less sensitive to the heterogeneous model employed than those obtained using ICE1, as a result of the distinct spatial distributions of the two ice models with respect to the anomalous mantle structures. As an exception to the general trend, in the Mediterranean region ICE1 appears to perform better than ICE3G for the FE models considered. It is known that a difference between the predictions based upon the two chronologies is evident for the sites of southern France, as recently discussed by Stocchi et al. (25). However, when all of the Mediterranean sites are considered simultaneously, as done here, the improvement of ICE1 with respect to ICE3G is not found to be statistically significant. (ii) In some of the regions considered in Fig. 9 [namely, the Hudson Bay, the Baltic (centre), the NW Territories and the Baltic (margins)], when ICE3G is employed, models and perform equally well, and the same holds for and. This indicates that, once the 3-D structure of the lithosphere is given, the introduction of the (3SMAC-based) heterogeneities in the shallow upper mantle does not improve significantly the agreement with the observations from these regions. In the case of the Hudson Bay and of the Baltic (centre) regions, our findings may appear at odds with published results showing that RSL data near the centre of the rebound are slightly more sensitive to lateral variations in the asthenosphere than to variations of the lithospheric thickness. This is true for parabolic ice sheets with size comparable to Fennoscandia (Sabadini et al. 1986; Gasperini & Sabadini 1989; Kaufmann et al. 1997) or Laurentia (Wu & van der Wal 23), or for more realistic ICE3G models (Kaufmann & Wu 22). However, none of the mentioned studies deal with the same 3-D heterogeneities (in terms of geometry and viscosity values) that we have employed here (see Fig. 3). This shows that the sensitivity of RSL data to mantle heterogeneities critically depends on the model assumptions. Even if we have not performed any search in the parameter space of our heterogeneous models, it is plausible that different viscosity values and shape of the heterogeneities could significantly change the results of Fig. 9. We have verified that the misfits of UL and VL do not differ at the 95 per cent confidence level except at the margins of the Baltic region, where the data can be better explained by heterogeneous models including laterally varying lithospheric thickness. This sensitivity to thickness of the lithosphere, which could only be guessed from the site-by-site analysis of Section 3.2, basically confirms previous results based on 2-D numerical models (see e.g. Kaufmann et al. 1997). When we turn our attention to the central portion of this region (see frame Baltic, centre), where a relatively small number of post-glacial RSL observations are available (namely 3), we do not observe any statistically significant differences in the regional misfits obtained with the four FE models. In this specific case, we explicitly observe how the choice of the ice aggregate may affect the agreement with the data, with model ICE1 (circles) showing a 6 Hudson Bay (11-18, N s = 6) Baltic (center) ( , N s = 3) 6 NW Territories (19-159, N s = 4) 6 Atlantic coast (31-359, N s = 46) Baltic (margin) (21-231/ , N s = 23) (e) 2 Pacific Islands ( , N s = 23) (f) 2 Mediterranean ( , N s = 7) (g) 6 British Islands ( , N s = 13) (h) Figure 9. Regional misfits M r for eight selected regions. Circles and crosses indicate the misfits between RSL observations and predictions when the deglaciation chronologies of ICE1 and ICE3G are employed, respectively. The labels on the x-axis indicate the forward model employed (see Fig. 2). shows the results obtained when the self-consistent is employed. C 26 The Authors, GJI, 165, Journal compilation C 26 RAS

10 GIA and RSL in a spherical 3-D earth Global (η LM ) (11-682, N s = 312) Global (η LM 1 22 ) (11-682, N s = 312) Figure 1. Global misfit computed using all of the data from the TP database. As in Fig. 9, the x-axis reports the models adopted and with crosses and circles we show predictions based on ICE3G and ICE1, respectively. In frame the lower mantle viscosity is Pa s, whereas in it has been increased to 1 22 Pa s. dramatic (and statistically significant) variance reduction with respect to UL when a 3-D lithosphere is introduced (VL models). As reported above, this is not observed when ICE3 is employed. (iii) From visual inspection of Fig. 9, we observe that the regional misfits generally tend to decrease when the is employed to predict RSL variations using the ICE3G chronology. The variance reduction of with respect to is statistically significant (at the 95 per cent confidence level) for the Hudson Bay region, for both the central and the peripheral portions of the Baltic region, and for the NW Territories. It is noteworthy that the allows for a variance reduction with respect to even in the centre of the formerly glaciated areas, to confirm that the process of hydroisostasy plays a role even during the post-glacial period. These findings confirm that the implementation of the gravitationally selfconsistent is needed to obtain fully reliable RSL predictions by spherically symmetric models, before addressing the problem of mantle viscosity inferences from GIA observations (Mitrovica & Peltier 1991; Wu 24; Spada & Stocchi 26). However, it is also apparent that the variance reduction obtained introducing the is generally significantly smaller than that observed modifying the ice-sheets time history. Since our code does not solve for the in a 3-D Earth, we must await further modelling improvements before drawing any definitive conclusion about the sensitivity of these RSL data to the heterogeneous structures introduced in our modelling. Results based on a restricted subset of the Hudson Bay RSL observations and on mantle heterogeneities of different geometry (Wu et al. 25) have already suggested that the 3-D structures can indeed be resolved using RSL data near the centre of the rebound. (iv) The important issue of the global effects of laterally varying lithospheric thickness and mantle viscosity on RSL is addressed in Fig. 1, where the regional misfit has been computed for all of the RSL sites contained in the database of Tushingham & Peltier (1993). In Fig. 1a, the lower mantle viscosity is Pa s, the value that we have used throughout the manuscript, while in Fig. 1b it has been increased to 1 22 Pa s, a value more consistent with the results of Lambeck and coworkers (see e.g. Lambeck (1995) and references therein), who have based their studies on ice-sheets chronologies that differ from those adopted here. The diagrams show that, regardless of the ice-sheets chronology and lower mantle viscosity employed, the misfits pertaining to the heterogeneous models do not differ significantly (at the 95 per cent level) from those relative to. This finding, which poses serious questions about the usefulness of global RSL data to constrain the heterogeneous structure of the mantle, results from a mutual cancellation of the effects of lateral variations on a global scale, which concur to RSL as a perturbation with a vanishing average. Since this result is basically unaffected by the choice of the ice-sheets chronology and of the lower mantle viscosity, this is likely to mainly depend on the spatial distribution of the RSL sites and to their location with respect to the heterogeneous structures of our 3-D model. Of course, as indicated by Fig. 9 and by a number of previous investigations, the role of lateral viscosity variations are relevant on a regional scale, and even more when the RSL curves are studied site by site, as done in Section 3 above. However, from the regional and local investigations it is also clear that when we are looking at real data, the effects of lateral viscosity variations cannot be isolated from those due to the surface load, as we have shown here comparing the outcomes of the two global ice chronologies ICE1 and ICE3G. ACKNOWLEDGMENTS We thank Patrick Wu, Ondrej Čadek and Rachel Cassidy for checking the manuscript, and an anonymous reviewer for their very constructive comments. 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