Geophysical Journal International Geophys. J. Int. (2011) 186, 1036–1044 doi: 10.1111/j.1365-246X.2011.05116.x

Ongoing glacial isostatic contributions to observations of sea level change

Mark E. Tamisiea National Oceanography Centre, Joseph Proudman Building, 6 Brownlow Street, Liverpool, L3 5DA, UK. E-mail: [email protected]

Accepted 2011 June 15. Received 2011 May 30; in original form 2010 September 8 Downloaded from https://academic.oup.com/gji/article/186/3/1036/589371 by guest on 28 September 2021 SUMMARY

Studies determining the contribution of water fluxes to sea level rise typically remove the ongoing effects of glacial isostatic adjustment (GIA). Unfortunately, use of inconsistent ter- minology between various disciplines has caused confusion as to how contributions from GIA should be removed from altimetry and GRACE measurements. In this paper, we review the physics of the GIA corrections applicable to these measurements and discuss the differing nomenclature between the GIA literature and other studies of sea level change. We then ex- amine a range of estimates for the GIA contribution derived by varying the Earth and ice models employed in the prediction. We find, similar to early studies, that GIA produces a small (compared to the observed value) but systematic contribution to the altimetry estimates, with a maximum range of −0.15 to −0.5 mm yr−1. Moreover, we also find that the GIA contri- bution to the mass change measured by GRACE over the ocean is significant. In this regard, we demonstrate that confusion in nomenclature between the terms ‘absolute sea level’ and

GJI Gravity, geodesy and tides ‘geoid’ has led to an overestimation of this contribution in some previous studies. A component of this overestimation is the incorrect inclusion of the direct effect of the contemporaneous perturbations of the rotation vector, which leads to a factor of ∼two larger value of the degree two, order one spherical harmonic component of the model results. Aside from this confusion, uncertainties in Earth model structure and ice sheet history yield a spread of up to 1.4 mm yr−1 in the estimates of this contribution. However, even if the ice and Earth models were perfectly known, the processing techniques used in GRACE data analysis can introduce variations of up to 0.4 mm yr−1. Thus, we conclude that a single-valued ‘GIA correction’ is not appropriate for sea level studies based on gravity data; each study must estimate a bound on the GIA correction consistent with the adopted data-analysis scheme. Key words: Satellite geodesy; Sea level change; Time variable gravity.

to the Earth model employed in the analysis, and the plausible range 1 INTRODUCTION of values has not been established. A number of recent studies have investigated the sea level budget The impact of GIA on the estimates of ocean mass change derived using three complementary measurements: altimetry (e.g. Jason- from GRACE has caused much debate. Indeed, ocean mass balance 1), gravity (Gravity Recovery and Climate Experiment, GRACE) studies have used estimates near −1mmyr−1 (Willis et al. 2008; and thermosteric variations (Argo) (Willis et al. 2008; Cazenave Leuliette & Miller 2009) or −2mmyr−1 (Cazenave et al. 2009). et al. 2009; Leuliette & Miller 2009). The ongoing effects of glacial These values are based on GIA model predictions developed by isostatic adjustment (GIA), that is, the continuing response of the Paulson et al. (2007) and Peltier (2004), respectively. The significant viscoelastic Earth to the loading from the ice age, impact the first discrepancy in the value adopted in previous studies is surprising two of these measurements. In this regard, the contribution of GIA given that the GIA predictions were both derived using the ICE- to altimetry is generally cited as −0.3 mm yr−1 (thus, subtracting 5G ice model with some form of VM2 Earth model (Peltier 2004; the GIA contribution implies adding 0.3 mm yr−1 to the observed Paulson et al. 2007). Peltier (2009) derived a value of −1.8 mm yr−1 altimetry rate), following the value derived by Peltier (2001). Peltier from the ICE-5G(VM2) (Peltier 2004) and explored the sensitivity (2009) found same value based upon ICE-5G(VM2) (Peltier 2004), of this estimate to different smoothing values and exclusion of but a slightly more negative value of −0.32 mm yr−1 when averaging particular spherical harmonic components of the model prediction. over a reduced latitude range of ± 66◦.However,thisrateissensitive It is interesting to note that several of the studies (Leuliette & Miller

1036 C 2011 National Environment Research Council (NERC) Geophysical Journal International C 2011 RAS Isostatic contributions to sea level change 1037

2009; Cazenave et al. 2009; Peltier 2009) have claimed closure of Classically, GIA models have focused on predictions of sea level the sea level budget using the three observation techniques despite because many of the time-series used as constraints in GIA mod- using these significantly different estimates of the GIA contribution elling are from (broadly defined) paleoshoreline data. [A compre- to the mass estimate derived from GRACE. In the effort to better hensive discussion of the general concepts involved in the prediction constrain the mass flux into the oceans, it is important to understand of GIA-induced sea level changes, and the first modern theoretical the range of uncertainty in the GIA contribution to the GRACE treatment of these changes, may be found in the canonical work of observation. Farrell & Clark (1976).] Given the relatively long time scale of GIA, This paper addresses three issues. First, GIA studies of sea level sea level variations driven by this process are predicted under the often use the term geoid interchangeably with absolute sea level or assumption that the evolving ocean is in static equilibrium (surfaces sea surface. We begin by reviewing these GIA calculations to rig- of constant pressure and density are equipotentials). This static sea orously describe the physical meaning of the predicted quantities. level theory treats the sea surface as an equipotential surface (i.e. This discussion clarifies the GIA contribution to ongoing changes no dynamic effects are taken into account). However, it is impor- in sea level as measured by either altimetry and gravity missions. tant to note that the value of the potential that defines the surface This review also demonstrates that a GIA correction to GRACE will be time dependent (e.g. Dahlen 1976; Farrell & Clark 1976), estimates of ocean mass balance based upon absolute sea level pre- as becomes evident when one considers that sea level was over Downloaded from https://academic.oup.com/gji/article/186/3/1036/589371 by guest on 28 September 2021 dictions (e.g. Peltier 2004) is inconsistent with the observation it is 120 m lower at the Last Glacial Maximum. It is the time dependence correcting and is thus in error. Secondly, for both altimetry and grav- of the potential value that has lead to confusion of terminology in the ity observations we estimate a plausible range of values associated past. Note that we also assume the density of water is constant, both with uncertainties in Earth and ice sheet models. Understanding spatially and temporally. These assumptions will hold throughout the uncertainty in these predicted contributions is vital to assessing the paper. the constraints imposed by the altimetry and gravity observations. The sea level predictions in GIA literature are typically a measure Finally, we illustrate that the GIA correction to a GRACE esti- of ‘relative sea level’, SR(θ, φ, tj), which is a globally defined field mate of ocean mass change will vary significantly depending upon at co-latitude θ, east-longitude φ and time tj. As an example, if the analysis techniques and averaging regions applied to the data. tide gauges could be deployed globally and were only affected by We will conclude that a universally applicable ‘GIA correction’ to GIA, they would observe S˙ R (θ,φ,tp), where the dot indicates a mass change measurements over the ocean is neither possible nor time derivative and tp is the present-day time (see Fig. 1a). The appropriate. ocean is bounded by two surfaces, the sea surface and the crust, and relative sea level refers difference between these two boundaries (see Fig. 1b)

2 TERMINOLOGY SR (θ,φ,t j ) = SA(θ,φ,t j ) − R(θ,φ,t j ). (1)

Studies of global sea level aim to quantify changes to both the total In this equation, SA(θ, φ, tj) is the absolute sea level or sea surface ocean volume and mass. If the Earth was rigid and the observa- and R(θ, φ, tj) is the height of the solid surface, assuming that these tions were made in a well-realized reference frame, then altimetry are measured relative to a common datum (e.g. the centre of mass would measure changes in the sea surface (or ocean volume), while of the Earth system, CM). Examples of S˙ A(θ,φ,tp)andR˙ (θ,φ,tp) GRACE would measure changes in ocean mass. In practise, nei- are shown in Figs 1(c) and (d). As indicated by the time derivative ther of these conditions is met. However, part of Earth’s non-rigid of eq. (1), in Fig. 1 the panels (a) = (c) − (d) The shoreline at t character is frequently incorporated into such studies. For example, = tj is a location where SR(θ, φ, tj) = 0, or, alternatively, where the GRACE observations of geopotential change are generally reported height of the sea and solid surfaces are the same. in terms of change in equivalent water height (EWH) (e.g. Wahr In GIA models, the solution to eq. (1) is obtained using the sea et al. 1998, eq. 14). This conversion provides estimates of the level equation introduced by Farrell & Clark (1976). The central geopotential change in EWH as a result of changes in the surface principle to these solutions is that the volume of water in the Earth’s load (from water and atmospheric pressure) at the Earth’s surface, system must be conserved. Thus, the change in SR between times after allowing for the Earth’s elastic response to the changing sur- t j−1 and tj integrated over the oceans must be equal to water that face load. entered or left the oceans during that time period. The only way Beyond this elastic response of the Earth to the changing distribu- that this conservation of mass can be achieved is if the sea surface, tion of the surface load around the world, GIA causes a response in the upper boundary of the ocean, moves from one equipotential to both the solid Earth and the ocean. In particular, continuing defor- another. For GIA at present time, we assume that there is no water mation and flow of the crust and mantle beneath the ocean changes currently being exchanged between the oceans and the ice sheets, both the seafloor and the equipotential in the region. Note that be- and Fig. 1(a) sums to zero. cause the mass change associated with GIA in oceanic regions is Return to Fig. 1(b) for an illustration. Wedenote the equipotential primarily due to the flow in the Earth, which has a much greater associated with the sea surface at t = t j−1 by (t j−1), and the density than water, the conversion of geopotential change to EWH perturbation to the potential, from t j−1 to tj, on this surface as greatly amplifies the size of the change. To account for the GIA δ(θ, φ, tj). Following equilibrium theory, the change in absolute contribution to both of these measurements, predictions of both sea sea level across the time interval is given by surface and geoid change are needed. However, as we will describe δSA(θ,φ,t j ) ≡ SA(θ,φ,t j ) − SA(θ,φ,t j−1) below, the GIA literature that uses the sea level equation frequently δ(θ,φ,t ) (t ) interchanges the terms ‘absolute sea level’ and ‘geoid’; a problem = j + j , (2) that is highlighted when comparing and discussing predicted GIA g g contributions to altimetry and GRACE observations. We review where g is the surface gravitational acceleration. The first term the relationship between ‘absolute sea level’ and ‘geoid’ in this on the right-hand-side of eq. (2) represents the shift in the radial context. position of the original equipotential (t j−1) from t j−1 to tj, and the

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Figure 1. Example model predictions of the ongoing GIA contributions to geodetic measurements, showing the (a) relative sea level change (tide gauges), (c) absolute sea level change (altimetry), (d) crustal motion (GNSS receivers), (e) equipotential change observed on Earth, (f) equipotential change observed by GRACE, (g) change in equivalent water height (GRACE). These examples use ICE-5G and VM2 (Peltier 2004) as input models. (b) Cartoon illustrating the crust and sea surface at an initial time (tj−1, solid black line) and a later time (tj, solid red line). The dotted red line illustrates the new position at time tj of the potential surface that corresponded to the sea surface at time tj−1. (h) Equivalent to (g), but for a Earth model with a lithospheric thickness of of 120 km, νUM 21 21 = 0.8 × 10 Pa s, and νLM = 3 × 10 Pa s. new position of (t j−1)attj is represented by the dotted red line such as the ocean, are reflected in this term. However, taking the in Fig. 1(b). This term includes effects due to changes in mass average of this term over the Earth must equal zero because the Earth distribution and changes in the centrifugal potential resulting from system did not lose or gain mass. For our present-day example, the changes in Earth rotation. All changes in mass in a given region, time derivative of δ(θ, φ, tp) is shown in Fig. 1(e).

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The spatially invariant second term of eq. (2) is included because, The impact of removing this term is illustrated by the difference as described earlier, the sea surface does not have to remain on the between Figs 1(e) and (f). The remainder of the discrepancy be- same equipotential as a function of time [i.e. (tj) = (tj) − tween estimates based on GIA predictions from Peltier (2004) and (t j−1)](Dahlen 1976). The integrated change of δ(θ, φ, tj)and Paulson et al. (2007), after accounting for the (tp)/g term de- δR(θ, φ, tj) over the oceans may not equal the change in volume scribed above, is due to differences in the degree two, order one associated with water entering or leaving the ocean. Thus, the sea terms (Chambers et al. 2010). While this factor of slightly greater surface must shift to a different potential value, and this shift is than two described in Appendix explains part of discrepancy, the represented by (tj). In other words, assuming a constant density reason for the remaining difference is unknown and thought to be of water, the change in mass of the water in the ocean must be an error in this component of δSA of Peltier (2004; Chambers et al. balanced by the change in the difference between the sea surface 2010). An appropriately-modified prediction of the δ(θ, φ, tj)/g height and crustal height over the ocean. The final position of the sea will be represented by δSG(θ, φ, tj) in the discussion below. surface is illustrated in Fig. 1(b) by the solid red line. Rather than a change in potential due to changes in mass, which is completely 3 MODELLING represented by δ(θ, φ, tj),  (tj) is simply introduced to allow us to track the changes of the sea surface with time. The difference To derive estimates of the current changes in relative sea level and Downloaded from https://academic.oup.com/gji/article/186/3/1036/589371 by guest on 28 September 2021 between Figs 1(c) and (e) is simply the spatially constant value its two bounding surfaces from eq. (1), one can solve the ‘sea level (tp)/δt/g. equation’ which describes how these surfaces are related and change GIA nomenclature that utilizes the sea level equation has fre- with time. In this regard, the equilibrium theory we adopt is revised quently equated the change in the sea surface height, the change in from the Farrell & Clark (1976) approach to include the influence the absolute sea level, and the change in the geoid height. Indeed, of Earth rotation changes as well as a coastline that evolves due this assumed equivalence is emphasized by the fact that, in many to either relative sea level changes or variations in the extent of cases, SA is often referred to as G. However, if the change in po- grounded, marine-based ice (e.g. Mitrovica et al. 2005; Kendall tential surface representing the geoid had a spatially constant term, et al. 2005). This theory will generate predictions of the present-day this would represent a change in mass of the earth system. There- (t = tp) secular change of absolute sea level, the radial position of fore, it is important in geodetic analysis of GIA model predictions the solid surface, and relative sea level due to GIA [i.e. δSA(θ, φ, t), that a clear distinction is made between the change in a particular δR (θ, φ, t)andδSR(θ, φ, t)dividedbyδt]. We denote these fields potential surface, δ(θ, φ, tj)/g, and the change in sea surface, by S˙ A(θ,φ,tp), R˙ (θ,φ,tp)andS˙ R (θ,φ,tp), respectively (from now δSA(θ, φ, tj). The distinction becomes clear when interpreting the on, the θ, φ, tp dependence of these quantities will be assumed). GIA prediction in the context of measurements made by Jason and The sea level theory requires two inputs: a global model of the GRACE. Pleistocene history of ice sheet change, and a model for the Earth’s Altimeters measure the sea surface height, and thus sample δSA. viscoelastic and density structure. For the former, we adopt a ver- However, GRACE is only sensitive to changes in mass distribu- sion of the ICE-5G deglaciation history (Peltier 2004). Because our tion and not . Therefore, one must not use the absolute sea purpose here is to present predictions of large-scale spatial aver- level predictions in place of geoid predictions when removing GIA ages of the absolute and relative sea level rates, our main concern estimates from GRACE observations. As an example of the differ- in choosing an ice history is that it be roughly consistent with the ence between these two predictions, the time derivative of the term total volume of grounded ice thought to have melted from the last −1 (tp)/g can range from −0.1 to −0.4 mm yr for the models glacial maximum to the end of the main deglaciation phase (4 kyr considered below. Assume a value of −0.2 mm yr−1. If one were to BP). We will briefly discuss comparative results based on ICE-3G include this spatially constant (spherical harmonic degree 0) term (Peltier 1994). in to the conversion of geopotential change to EWH (in this case, The GIA predictions of present day rates of relative sea level multiplying by the ratio of the average density of the Earth to wa- and its bounding surfaces are sensitive to variations in the adopted ter divided by three; e.g. eq. 14 of Wahr et al. 1998), this term Maxwell viscoelastic Earth model. In this regard, we have generated would add −0.37 mm yr−1 EWH to the predicted GIA contribution results for a suite of spherically symmetric, self-gravitating, com- to the GRACE measurement. Failure to recognize this distinction by pressible Earth models distinguished on the basis of the thickness Peltier (2009) accounts for roughly half of the discrepancy between of a purely elastic lithosphere (LT) and assumed constant viscos- his estimate of the GIA contribution to GRACE observations and ity within the sublithospheric upper (νUM) and the lower (νLM) 20 21 the estimate derived from Paulson et al. (2007) (Chambers et al. mantle: 71 km < LT < 120 km, 10 <νUM < 10 Pa s, and 21 22 2010). 2 × 10 <νLM < 5 × 10 Pa s. The boundary between these Using δSA predictions directly in GRACE analyses can lead to latter two regions is taken to be 670 km depth. The elastic and another error if the impact of contemporaneous perturbations in the density structure of the Earth model is prescribed by the seismic rotation vector on sea level (i.e. rotational feedback) are included. model PREM (Dziewonski & Anderson 1981). Results are pre- Most recent GIA predictions are computed for an observer on the sented in the centre of Earth (CM) reference frame. GIA calculations rotating Earth, and they therefore include the full contribution due are usually conducted in the centre of solid Earth (CE) reference to polar motion. However, the direct component of this contribution frame (e.g. Farrell 1972). However, the difference between GIA (i.e. the shift in the orientation of the centrifugal potential) would predictions in the CE and CM is small because this difference is not be present in the GRACE data, because the observation is not due only to the ongoing changes in the ocean water distribution due made in the rotating frame. Therefore the direct component must to GIA (Argus 2007; Klemann & Martinec 2009). (Note, the same be removed when calculating the GIA correction to a GRACE ob- statement could not be made for the large present-day changes in ice servation (e.g. one must remove the value of ‘1’ multiplying the volumes.) δ-function in eq. (A11) of Mitrovica et al. (2001)). Removing It should be noted that an ice sheet model used in the sea this term causes the size of the degree two, order one coefficient to level equation is derived using a specified Earth model, VM2 be reduced by a factor of slightly greater than two; see Appendix. in the case of ICE-5G (Peltier 2004). Varying the Earth model

C 2011 National Environment Research Council (NERC), GJI, 186, 1036–1044 Geophysical Journal International C 2011 RAS 1040 M. E. Tamisiea independently of the ice sheet model may introduce larger varia- described above. For this plot, the lithospheric thickness is chosen tions in the GIA estimates than would result if the Earth model to be 71 km. This choice of LT has a small impact on these results: and ice loading history were varied simultaneously. However, our generally less than a 15 per cent change, with a thicker lithosphere suite of predictions will nevertheless provide an indication of the producing smaller predictions for an upper-mantle viscosity above sensitivity of the GIA predictions to variations in the Earth model. 3 × 1020 Pa s. We will discuss in more detail the sensitivity of the In addition, the Earth is characterised by 3-D variations in man- predictions to variations in lithospheric thickness when reviewing tle properties, and the uncertainties caused by the neglect of these the GIA contribution to GRACE observations. variations cannot be adequately captured using a suite of 1-D pro- As indicated by eq. (1), the third column in Fig. 2 represents the files. Thus, while the range of model predictions we investigate difference between the first two columns. Because the global average is relatively large, narrowing the range of possible values without of S˙ R over the ocean is equal to the change in its water volume, the consideration of the other sources of uncertainty is premature. present-day values for this plot are zero for GIA, and the plots for S˙ A and R˙ are identical. However, if one restricts the average to a smaller region of the ocean, S˙ would be non-zero. Most of the GIA-induced 4 RESULTS R vertical deformation of the crust in the ocean occurs in high latitudes In this section, we start by examining the GIA contribution to sea near the loading centres (e.g. Mitrovica & Peltier 1991). The impact Downloaded from https://academic.oup.com/gji/article/186/3/1036/589371 by guest on 28 September 2021 level change as measured by satellite altimetry, S˙ A. We also will plot of ongoing GIA on oceanic volume is mainly due to the collapse of the predictions for R˙ and S˙ R to better illustrate the physical mecha- the forebulge near these regions. Thus, as an extreme example, one nisms responsible for the corrections. Then, we consider measure- could limit averages to the mid-latitudes. In this case the magnitudes ments of sea level variations over the ocean as measured by GRACE for the values of R˙ would be significantly smaller and the average and show that GIA is responsible for a significant contribution due of S˙ R would be negative as water is being transferred to higher to the nature of the measurement. latitudes. Estimates of secular ‘global sea level change’ inferred from The main result in Fig. 2 is the predicted average of the GIA- satellite-derived data sets (e.g. TOPEX/Poseidon, Jason-1 and induced absolute sea level rate over regions sampled by the altime- Jason-2) are derived from the rate of change of the absolute sea try missions (bottom of first column). This average ranges from level averaged over some large geographic area. In the case of these ∼−0.15 to −0.5 mm yr−1 for the Earth models we have con- missions, this average is obtained across oceanic regions bounded sidered. Thus, the mean (altimetry-averaged) absolute sea level to the north by 66◦N and to the south by 66◦S, excluding Hudson rate obtained from altimetry data could be increased by 0.15 to Bay and areas with ocean bathymetry less than 120 m (in the case of 0.5 mm yr−1 by a correction for the signal due to GIA. We note that Leuliette et al. 2004). A correction of this estimate for GIA should this correction remains relatively constant as one considers averages be obtained by the removal of a similarly averaged map of absolute over a larger subregion of the ocean. Thus our conclusion will hold sea level rates (Peltier 2001). for satellite-derived estimates with a sampling range that is different Fig. 2 shows the average values of S˙ A, R˙ and S˙ R for two cases: from the current Jason-type missions. It should be noted that there the global ocean and the ocean sampled by the altimetry missions as is only a small change in the results if ICE-3G is used instead of

Figure 2. Comparison of GIA contribution to altimetry estimates of sea level rise. The contours, with units of mm yr−1, show the variation caused by changing the upper- or lower-mantle viscosity. All plots assume a lithospheric thickness of 71 km. The first column shows the altimetry contribution from GIA, the second column shows the average change of crustal depth over the ocean, and the final column shows the difference between the first two. The results in the first row are generated by averaging over the global ocean, and the second restricts the average to latitudes between ± 66◦.

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Figure 3. Comparison of GIA contribution to GRACE estimates of sea level change over the ocean. The contours, with units of mm yr−1,showhowthe water equivalent average varies with upper- and lower-mantle viscosity. The rows assume a lithospheric thickness of 71 and 120 km, respectively. The left column uses the present-day ocean mask, truncated to degree and order 60, from the GIA prediction; the middle column uses the same mask with an additional 500 km Gaussian smoothing; and the final column uses a non-optimised version of the mask employed by Leuliette & Miller (2009).

ICE-5G, with ICE-3G producing slightly larger values, peaking at (2009). The two rows in Fig. 3 show the results for the limiting 21 ∼20 per cent larger for νLM = 2 × 10 Pa s. values of lithospheric thickness. Next, we consider the GIA contribution to the GRACE obser- The variation between filters shown in Fig. 3 can be explained vation of geoid height change. The GIA estimate that should be by considering the regions excluded. For example, B&R = 300 ◦ compared to the GRACE data differs from the S˙ A in two signifi- excludes the latitudes greater than 66 and regions within 300 km cant ways. First, the geoid does not include the spatially invariant of the continent. Thus, it excludes many of the areas experiencing second term in eq. (2) or the direct effect of the perturbed cen- mass increase, such as Hudson Bay and the Antarctic ice shelves trifugal potential (e.g. Fig. 1f). Secondly, most GRACE results that would be included in the R = 0 mask, and this yields a much are quoted in terms of ‘equivalent water height’, the change in more negative value. The decrease in magnitude from the R = 0 height of water that would cause the observed geopotential change to the R = 500 masks is due to the opposite effect. The smoothing on an elastic Earth. As discussed in Section 2, because the varia- effectively includes more of these regions of mass increase, thus tions caused by GIA are due primarily to crustal motion a relatively lowering the average magnitude. small height variation will produce significantly larger apparent Generally, the estimated GIA contribution varies most strongly changes in EWH (e.g. Fig. 1g). Thus, the GIA contribution be- with lower mantle viscosity. For the mid-viscosity range, 8 × 1021 22 21 comes a significant component in the global water balance estimates Pa s ≥ νLM ≤ 2 × 10 Pa s for νUM = 10 Pa s, the periph- when reconciling total regional contributions to observations (Willis eral bulge surrounding Laurentia extends further into the ocean et al. 2008; Cazenave et al. 2009; Leuliette & Miller 2009; Peltier and the Earth has recovered less than for lower values of νLM. 2009). Thus, the average over the ocean is more negative. For the pur- Fig. 3 shows the variation of the GIA contribution to an ocean- pose of discussion, assume a lithospheric thickness of 71 km and 20 averaged value of S˙G , with contour lines representing change in an upper-mantle viscosity of 5 × 10 Pas.Then,byvaryingthe mm yr−1 EWH, obtained in three ways. The first column assumes lower mantle viscosity, the resulting averages range from −0.82 to an average using the present-day ocean mask derived in the GIA −1.72 mm yr−1 using the ocean mask with no smoothing, and −0.86 calculations but truncated at spherical harmonic degree and order to −2.10 mm yr−1 for the B&R = 300 mask. As would be expected 60. (Truncation at degree 60 has a negligible impact on the aver- when the results are dominated by lower mantle viscosity, the lower- ages; we refer to this mask as R = 0.) The second column applies a valued spherical harmonics contribute significantly. In particular, 500 km Gaussian smoothing to the mask (R = 500), a common pro- the degree 2 contribution is significantly enhanced in the B&R = cessing step applied to GRACE data to reduce the impact on mass 300 mask (between −0.3 and −0.7 mm yr−1) when the higher lati- estimates of increasing error with degree (Wahr et al. 1998). How- tudes are excluded from the average. ever, with GRACE data, there is considerable leakage of the much One can draw two conclusions from these results. First, uncer- larger gravity variations over the continents into the ocean averages tainty in Earth model could have a significant impact on the esti- if a border around the continents is not maintained (e.g. Chambers mated contribution that would be observed by GRACE. As a visual 2009). Thus, the third column uses a mask (B&R = 300) that main- example, Figs 1(g) and (h) show that even switching from VM2 tains a 300 km border around the continents, limits the averaging to an Earth model that provided good results in Tamisiea et al. to between ± 66◦, and applies a 300 km Gaussian smoothing. This (2007) can produce significant changes to the average, −0.77 to mask is a non-optimized version of that used in Leuliette & Miller −0.97 mm yr−1. Secondly even if the GIA prediction were known

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Figure 4. Comparison of GIA contribution to GRACE estimates of sea level change over the ocean. The contours, with units of mm yr−1,showhowthe water equivalent average varies with upper- and lower-mantle viscosity. The results in each row were generated using either ICE-5G (Peltier 2004) or ICE-3G (Tushingham & Peltier 1991). The columns are described in Fig. 3. The lithospheric thickness for all of the results is 71 km. with absolute certainty, the different averaging regions and pro- predictions. For ICE-3G and an Earth model with LT = 120 km, 21 21 cessing techniques applied to the GRACE data could change the νUM = 1 × 10 Pa s, and νUM = 2 × 10 Pa s, we find −1.11, − estimate significantly. Thus, it is impossible to have a single-valued 0.84 and −1.21 mm yr−1. GIA ‘correction’ for GRACE observations, as each analysis could sample the GIA contribution differently. Fig. 4 shows how the averages change when ICE-3G (Tush- 5 CONCLUSIONS ingham & Peltier 1991) is used instead of ICE-5G. These two ice models are significantly different (ICE-3G has more ice in Efforts to estimate global mean sea level variations that might be Antarctica and less in the Northern Hemisphere), but were de- associated with recent climate require a correction for the signal due rived using somewhat similar Earth models. The most signifi- to ongoing GIA. However, the nature of this correction depends on cant differences between the results occur at higher values of the sea level signal being estimated. Although the GIA contribution νUM and lower values of νLM. In absolute terms, the differ- to both altimetry and GRACE estimates of sea level rise are related ences are generally less than 0.3 mm yr−1,or25percentofthe to changes in geopotential height, satellite altimetry (e.g. Jason) total. measures changes in the position of the absolute sea level (sea For all model predictions, the estimated contribution is negative. surface) while GRACE measures the total mass change over the So, for example, assume that GRACE observations observed no ocean. The sea surface change caused by the crustal motion due increase in ocean mass. The results from the first column of Fig. 3 to GIA is relatively small. However, when the geoid change is would imply anywhere from 0.5 to 1.9 mm yr−1 must be contributed expressed in terms of equivalent water height, the larger density from other sources, say melting glaciers or ice sheets, to offset the associated with crustal material causes the GIA contribution to GIA contribution to this estimate. If we were to limit the set of Earth become of equal magnitude to the expected variation due to water models to those preferred in Tamisiea et al. (2007) using ICE-5G, fluxes into the ocean. then the range estimated would be between −0.9 and −1.5 mm yr−1 In the past, confusion has resulted from the use of the term for the full ocean mask with no smoothing or between −1.1 and −1.8 ‘geoid’ employed in the GIA literature. Because the GIA theory mm yr−1 for B&R = 300. However, these estimates are strongly is mainly concerned with the sea surface and assumes equilibrium biased by the mantle structure under Hudson Bay, and much of the adjustment of the ocean, the equipotential surface corresponding to sea level change is driven by the collapse of the peripheral bulge in the sea surface is often referred to as the geoid height, though the the ocean near the loading centres. Thus, a different average mantle associated equipotential value will change with time. Thus, altime- viscosity may alter the estimated range of values. Therefore, it may try measurements are appropriately compared to these estimated be premature to limit the range further without explicit application values. In the case of the current GIA contribution, the non-zero of other data sets. global average of sea surface change is due to change in the ocean Finally, if one looks at only the Earth model (VM2) used to derive volume due to crustal motion. However, for the estimated GRACE ICE-5G, we find a values of −0.96, − 0.77 and −1.09 mm yr−1 contribution, this non-zero global contribution must not be included for the three filters shown in Fig. 3. The difference between these because it does not represent a change of mass in the ocean; doing estimates and the results of Chambers et al. (2010) using the Paulson so would greatly increase the estimated mass change that would be et al. (2007) model is partially due to the filters employed in the observed by GRACE. Improperly including this term has caused estimation procedure and partially due to the difference in the model some estimates (Cazenave et al. 2009; Peltier 2009) to be too large

C 2011 National Environment Research Council (NERC), GJI, 186, 1036–1044 Geophysical Journal International C 2011 RAS Isostatic contributions to sea level change 1043

(Chambers et al. 2010). In addition, we have demonstrated that the reevaluation from GRACE space gravimetry, satellite altimetry and Argo, degree two, order one coefficients of the sea surface, S˙ A, would be Global planet. Change, 65, 83–88. a factor of two larger than those of S˙G , which can also lead to an Chambers, D.P., 2009. Calculating trends from grace in the presence of overestimate. large changes in continental ice storage and ocean mass, Geophys. J. Int., Some altimetry data sets are limited to oceanic regions between 176(2), 415–419, doi:10.1111/j.1365-246X.2008.04012.x. ◦ Chambers, D.P.,Wahr, J., Tamisiea, M.E. & Nerem, R.S., 2010. Ocean mass ±66 latitude. Our predictions of the average absolute sea level rate from GRACE and glacial isostatic adjustment, J. geophys. Res., 115, − −1 − −1 induced by GIA range from 0.15 mm yr to 0.5 mm yr ,where B11415, doi:10.1029/2010JB007530. the range reflects the suite of Earth and ice models we considered. Dahlen, F.A., 1976. The passive influence of the oceans upon the rotation of We furthermore find that this range is not significantly altered if the earth, Geophys.J.R.astr.Soc.,46, 363–406. we consider averaging zones that are more or less confined in lati- Dziewonski, A. & Anderson, D., 1981. Preliminary reference earth model, tude. Our results for the GIA contribution to altimetry observations Phys. Earth planet. Inter., 25, 297–356. are in good agreement with predictions from earlier studies (Peltier Farrell, W.E.,1972. Deformation of the earth by surface loads, Rev. Geophys. 2001, 2009) generated using the same Earth and ice sheet models. Space Phys., 10, 761–797. The spread in our results demonstrate that the uncertainty in ice Farrell, W.E. & Clark, J.A., 1976. On postglacial sea level, Geophys. J. R. history and Earth structure will not lead to a significant increase in astr. Soc., 46, 647–667. Downloaded from https://academic.oup.com/gji/article/186/3/1036/589371 by guest on 28 September 2021 Kendall, R.A., Mitrovica, J.X. & Milne, G.A., 2005. On post-glacial sea the uncertainty of the global mean rate of S˙ . Current uncertain- A level – II. Numerical formulatioin and comparative results on spherically ties in the global mean are dominated by errors in the tide gauge symmetric models, Geophys. J. Int., 161, 679–706, doi:10.1111/j.1365- −1 calibration, and are at the level of 0.4 mm yr (Leuliette et al. 246X.2005.02553.x. 2004). Klemann, V. & Martinec, Z., 2009. Contribution of glacial-isostatic Unfortunately, the spread in the model predictions for the GIA adjustment to the geocenter motion, Tectonophysics, in press, contribution to the GRACE observations is quite large. Assuming doi:10.1016/j.tecto.2009.08.031. a straight average over the oceans (R = 0), a range of estimated Leuliette, E. & Miller, L., 2009. Closing the sea level rise budget produced by varying the Earth model was −0.5 to −1.9 mm yr−1, with altimetry, Argo, and GRACE, Geophys. Res. Lett., 36, L04608, while the difference between estimates calculated using the two doi:10.1029/2008GL036010. ice sheet models was up to 25 per cent. This range has significant Leuliette, E.W., Nerem, R.S. & Mitchum, G.T., 2004. Calibration of TOPEX/Poseidon and Jason altimeter data to construct a contin- implications for estimates of the present-day mass contribution to uous record of mean sea level change, Mar. Geod., 27, 79–94, sea level rise. If GRACE observed close-to-zero mass change over doi:10.1080/01490410490465193. the ocean, then this contribution would have to be compensated for Matsuyama, I., Mitrovica, J.X., Manga, M., Perron, J.T. & Richards, M.A., by mass loss from ice sheets, glaciers and water stored on land. 2006. Rotational stability of dynamic planets with elastic lithospheres, J. Given the importance of GRACE to help constrain the mass-flux Geophys. Res., 111(E2), E02003, doi:10.1029/2005JE002447. contribution to sea level rise, it becomes vitally important to narrow Mitrovica, J., Milne, G. & Davis, J., 2001. Glacial isostatic adjustment on a this range as much as possible. Restricting the Earth models to rotating earth, Geophy. J. Int., 174, 562–578. those preferred in Tamisiea et al. (2007) reduces this range to Mitrovica, J.X. & Peltier, W.R., 1991. On postglacial geoid subsidence over −0.9 to −1.5 mm yr−1. However, these predicted values are driven the equatorial oceans, J. geophys. Res., 96(B12), 20053–20071. by forebulge collapse in oceanic regions, and it is not clear that a Mitrovica, J.X., Wahr, J., Matsuyama, I. & Paulson, A., 2005. The ro- tational stability of an ice-age earth, Geophys. J. Int., 161, 491–506, restriction based on constraints derived from Laurentian adjustment doi:10.1111/j.1365-246X.2005.02609.x. are appropriate. Paulson, A., Zhong, S. & Wahr, J., 2007. Inference of mantle viscosity Even if the ice and Earth models are known perfectly, however, from grace and relative sea level data, Geophys. J. Int., 171, 497–508, our results indicate that a single-valued GIA ‘correction’ is not doi:10.1111/j.1365-246X.2007.03556.x. possible. The average value over the ocean can vary significantly Peltier, W.R., 1994. Ice age paleotopography, Science, 265, 195–201. depending upon the averaging area and processing techniques ap- Peltier, W.R., 2001. Global glacial isostatic adjustment and modern instru- plied to prediction. This variation can be as much as 0.4 mm yr−1. mental records of relative sea level history, in Sea Level Rise: History and Thus, it is important that each study process the GIA prediction in Consequences, Vol.75, pp. 65–95, eds Douglas, B.C., Kearney, M.S. & the same manner as the GRACE data to appreciate how GIA will Leatherman, S.P., Academic Press, San Diego, CA. contribute to their GRACE estimate. Peltier, W.R., 2004. Global glacial isostasy and the surface of the Ice-Age Earth: the ICE-5G(VM2) model and GRACE, Annu. Rev. Earth Planet. Sci., 32, 111–149. Peltier, W.R., 2009. Closure of the budget of global sea level rise over the ACKNOWLEDGMENTS GRACE era: the importance and magnitudes of the required corrections I would like to thank Chris Hughes, Jerry Mitrovica and Philip for global glacial isostatic adjustment, Quat. Sci. Rev., 28, 1658–1674, Woodworth for comments on the manuscript and Eric Leuliette doi:10.1016/j.quascirev.2009.04.004. for comments and providing his averaging mask used in GRACE Tamisiea, M.E., Mitrovica, J.X. & Davis, J.L., 2007. GRACE gravity data processing. Two anonymous reviewers provided suggestions that constrain ancient ice geometries and continental dynamics over Laurentia, Science, 316, 881–883, doi:10.1126/science.1137157. greatly improved the manuscript. This work was funded by NERC Tushingham, A.M. & Peltier, W.R., 1991. ICE-3G: a new global model as part of Oceans 2025. of late Pleistocene deglaciation based upon geophysical predictions of post-glacial relative sea level change, J. geophys. Res., 96, 4497– REFERENCES 4523. Wahr, J., Molenaar, M. & Bryan, F., 1998. Time variability of the earth’s Argus, D.F., 2007. Defining the translational velocity of the reference gravity field: hydrological and oceanic effects and their possible detection frame of Earth, Geophys. J. 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Now, if we add the direct affect of the perturbation to the rotational APPENDIX: RELATIONSHIP BETWEEN potential, ˙ ,toS˙ G to obtain S˙ A, while simultaneously subtracting THE DEGREE TWO, ORDER ONE the term to balance the equation, we have the expression COEFFICIENTS OF S˙ A AND S˙ G      A G A The degree two, order one coefficients of the S˙ and S˙ differ due m˙ 1 Re S˙ = Ma 10 21 to the fact that they are observed differently. (Note that we have − A m˙ 2 C A 3 −Im S˙ shifted the subscripts A and G to superscripts in the appendix for 21 clearer presentation when subscripts are added for the degree and    A ˙ order.) S˙ is observed on the Earth, and thus the direct effect of the Re 21 (A4) − Ma 10 . shift in the orientation of the centrifugal potential must be included. − g(C A) 3 −Im ˙ 21 Because S˙ G is observed from a non-rotating frame, this direct effect would not be present. It is possible to derive a relationship between From Mitrovica et al. (2001), we can relate ˙ to m˙ the degree two, order one coefficients for these two observations.     ˙ 2 2 Re 21 a m˙ 1 As is illustrated in Chambers et al. (2010), = √ . (A5)     Im ˙ 30 −m˙ Downloaded from https://academic.oup.com/gji/article/186/3/1036/589371 by guest on 28 September 2021 m˙ 1 I˙ 21 2 1 = 13 , (A1) m˙ 2 C − A I˙23 Thus, rearranging terms gives  where m˙ is change in the angular pole position, I˙ are the products of   Ma 10   ij ˙ A inertia, and C and A are the polar and equatorial moments of inertia. m˙ 1 − Re S21 = C A 3 , (A6) Using the normalization for the spherical harmonics presented in 1 A m˙ 2 + −Im S˙ ˙ 1 21 Matsuyama et al. (2006), Iij can be expressed as k f      G 5 2 ˙ Re S˙ where kf is the fluid Love number, given by (3 G/a / )(C − A). I13 = 10 21 , ˙ Ma (A2) Finally, degree two, order one coefficients of S˙ G and S˙ A can be I23 3 −Im S˙ G 21 related by m       where M is the Earth’s mass and a is the mean radius. Thus, ˙ can ˙ A ˙ G Re S21 1 Re S21 be expressed as = 1 + (A7) A G      Im S˙ k f Im S˙ ˙ G 21 21 m˙ Ma 10 Re S21 1 = . (A3) m˙ 2 C − A 3 − ˙ G Im S21 The factor in the parentheses above evaluates to 2.06.

C 2011 National Environment Research Council (NERC), GJI, 186, 1036–1044 Geophysical Journal International C 2011 RAS