Moho density contrast in Antarctica determined by satellite gravity and seismic models

showing how the main geological structures contribute to the MDC. Finally, we study the remaining glacial isostatic adjustment effect on gravity to ﬁgure out how much it affects the MDC recovery, yielding a correlation of the optimum spectral window (7 ≤ n ≤ 12) between XGM2019e and W12a GIA models of the order of ∼ 0.6 contributing within a negligible ± 14 kg m − 3 to the MDC.

The crust-mantle or Mohorovičić discontinuity (Moho) separates the Earth's crust from the upper mantle, with a density contrast, which traditionally was assumed to be of the order of 0.6 g cm −3 (e.g.Heiskanen & Moritz 1967, p. 135).Traditionally, the Moho depth (MD) is delineated from seismic data, such as active (controlledsource) seismic experiments using refracted or reflected seismic waves, passive methods based on seismic energy generated by earthquakes, receiver functions and the ambient noise technique (Laske et al. 2013 andSzwillus et al. 2019).The other Moho constituent, the Moho density contrast (crust-mantle density contrast or shortly MDC) can be estimated from the seismic signal velocity difference above and below the boundary converted to a density contrast.However, as seismic data collection is relatively costly, the data distribution is frequently sparse, in particular in Antarctica and over open seas.
With the advent of satellite gravity missions, it has been possible to image the MD and MDC via satellite gravity observations based on isostatic models.Such data allow more or less homogeneous coverage of the Earth by data (except for some restrictions in polar regions).
Although several authors have fairly well determined the MD by seismic or gravimetric techniques on global or regional scales, there are few studies specifically addressing the lithosphere density structure in Antarctica, where there is low spatial coverage of high-quality seismic data, which is the primary source of data for such studies.However, one can find some global MDC models derived from gravimetric or seismic experiments or combinations thereof.For instance, Sjöberg & Bagherbandi (2011) derived a method based on the Vening Meinesz-Moritz (VMM) technique and applied it globally in a model named KTH11, using gravity (EGM2008), topography (DTM2006) and seismic crust (CRUST2.0)data.Tenzer et al. (2012a) produced a global MDC 1952 C The Author(s) 2021.Published by Oxford University Press on behalf of The Royal Astronomical Society.This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/),which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited.9d) is rejected, that is, the two models are not consistent.RMS is root mean square of the difference between the two models.model from EGM2008, DTM2008, CRUST2.0 and ICE-5G GIA models, differing about 7% from the global average of the MDC estimated by Sjöberg & Bagherbandi (2011).Tenzer et al. (2012b) demonstrated that the MDC under the oceanic crust is highly spatially correlated with the ocean-floor age, and Tenzer et al. (2013) argued that the MDC varies globally from 81.5 kg m −3 in the Pacific region to 988 kg m −3 beneath the Tibetan Plateau.Reguzzoni et al. (2013) estimated a global Moho model, named GEMMA12C, from the combination of the CRUST2.0model and GOCE data, including both MD and MDC.Abrehdary et al. (2015) and ( 2017) presented two global Moho models using gravimetric data jointly with seismic data, containing both the MD and MDC and named them KTH14C and KTH15C.Abrehdary et al. (2019) estimated a new MDC model in ocean areas, using the marine gravity field from satellite altimetry in combination with a seismic crustal model and Earth's topographic/bathymetric data.The MDC can also be computed from the seismic crustal models CRUST2.0 (Bassin et al. 2000) and CRUST1.0(Laske et al. 2013) as the difference between the densities in the upper mantle and lower crust.
Attempts for determining specifically the Antarctic MDC by gravimetric and seismic methods have also been published by some authors, such as Tenzer & Bagherbandi (2013), who employed the VMM technique with gravimetric data and constraining information from the global seismic crustal model CRUST2.0.Unfortunately, the crustal depths modelled by CRUST2.0 are poor in Antarctica, as we demonstrate in Section 3.2.Also, Llubes et al. (2018) estimated the crustal thickness in Antarctica and used a mean MDC of 0.63 g/cm 3 when adjusting the crustal thickness based on adding satellite gravimetric data to seismological results.Unfortunately, the error in assuming a constant MDC will significantly affect the gravimetric-isostatic estimation of the MD; see eq. ( 5) below.Pappa et al. (2019a) combined seismological and petrological models with satellite gravity gradient data to obtain the thermal and compositional structure of the Antarctic lithosphere.They also calculated the Antarctic MD by gravity inversion and found lower MDC in East  Here we produce a new MDC model (named HVMDC20) in Antarctica and its surroundings based on the VMM technique.We take advantage of the seismic models of CRUST19 (Szwillus et al. 2019) and CRUST1.0, the global topographic model Earth2014 (Hirt & Rexer 2015) and the free-air gravity data provided by the new Earth Gravitational Model XGM2019e (Zingerle et al. 2019).The output of this model is presented in five steps.The first step is to determine the MDC, the second step is to model the MDC uncertainty, the third step is to compare our estimates for MDC with previous seismic and gravimetric models and the last ones are to investigate the impact of the delayed glacial isostatic adjustment (GIA) on gravity and discuss relevant geodynamic processes occurring in the area.

M E T H O D O L O G Y
In this section, the MDC estimation is presented as a gravimetricisostatic method.It implies that, the VMM problem is altered so that the Moho surface is known (e.g. from seismic data analysis), and the MDC is sought.

Vening Meinez-Moritz method
Our starting point is the following first-order global Fredholm integral equation for the MDC, denoted ρ, that satisfies the hypothesis of isostatic equilibrium, which yields a vanishing isostatic gravity disturbance δg I (P) for point P on the Earth's surface (cf.Sjöberg 2009Sjöberg , 2013)): where σ is the unit sphere, R is mean Earth radius, is the kernel function expressed by a series of Legendre's polynomials P n (cos ψ) with ψ being the geocentric angle between the integration point and computation point P, s = 1 − τ = 1 -D/R, D being the MD (at the integration point) and Here δg B (P) is the Bouguer gravity disturbance corrected for the gravitational contributions of topography/bathymetry and crustal density variations of the oceans, ice and sediments.A C0 is the compensation attraction for a 'normal' Moho surface (mean MD in the target region) at depth D 0 below sea level and G is the gravitational constant.As we will see below, neither the constant A C0 nor lower degree spherical harmonics of the gravity disturbance are used in the calculations, as they do not contribute to the isostatic balance.

Solution for a variable MDC by spherical harmonics
The left-hand side of the integral of eq. ( 1a) was expanded in spherical harmonics (Y nm ) to second order by Sjöberg & Bagherbandi (2011), yielding and by also expressing the right-hand side of eq. ( 1a) as a harmonic series one obtains the spectral equation where the bracket is the same as in eq. ( 2).By summing up the spectral components, one thus arrives at the following solution for the product D ρ: (5)   By further using the approximation ρ D 2 /R ≈ ρ D D 0 /R (see Eshagh 2017), where D 0 is the mean MD over the study area, one obtains after a few manipulations the following second-order solution for the MDC for a known MD and omitting terms of degree higher than n max : or (6b)

MDC uncertainty
Assuming there are no gross errors/blunders, the uncertainty of the MDC is composed of random and systematic errors, which can be expressed by the mean square error (Sjöberg & Bagherbandi 2011): where σ 2 ρ is the variance of the estimator of eq. ( 6a) or (6b) and the bias term consists of all contributions of systematic errors.As eqs (6a) and (6b) are based on the VMM method, it implies that any gravity signal that is not related with isostatic balance will cause a bias.Unfortunately, as no systematic error in this study is known, the only quality indicator we have in access is the standard error (STE), which is obtained by error propagation of the uncorrelated observation variances σ 2 f and σ 2 D (cf.Sjöberg & Bagherbandi 2011): As for the MD and its standard error provided by CRUST19, the problem is quite simple since the expected accuracy has already been computed by Szwillus et al. (2019), which we use in the present work.

N U M E R I C A L S T U D I E S
We will first describe the source of data to be used in the MDC determination (Section 3.1), followed in Section 3.2 by validation of CRUST1.0 versus CRUST2.0 and in Section 3.3 by the estimation of the MDC by assuming that the MD is known from the seismic Moho model CRUST19, and the section ends with estimating the MDC uncertainty.In Section 3.4, we compare the new MDC estimates with those of some other gravimetric-isostatic and seismic models (KTH11, GEMMA12C, CRUST1.0 and KTH15C).

Data sets
We employ the VMM model to image the MDC over Antarctica for both land and ocean areas.To reach this purpose, different data sets have been used, including the global Earth gravity field model (XGM2019e), the global topography model (Earth2014), and the global seismic crustal model (CRUST1.0 and CRUST19).In details, free-air gravity disturbances extracted from the XGM2019e global gravity field model in a set of 1 • × 1 • blocks over Antarctic and surrounding are used.XGM2019e is a combined global gravity field model represented through spherical harmonics, which we will use up to degree and order n max =180, corresponding to a spatial resolution of about 100 km, the approximate lower spatial limit for isostatic balance of the crust.The free-air gravity disturbances are applied to calculate the refined Bouguer gravity disturbance, including correcting the gravity data for known anomalous crustal density structures, mainly density variations of oceans, glacial ice, and sediments (i.e.stripping gravity corrections).For this purpose, the gravimetric forward modelling approach is used to compute the stripping gravity corrections using the ESCM180 Earth's spectral crustal model, which was derived based on refining the CRUST1.0global crustal model (cf.Tenzer et al. 2009Tenzer et al. , 2010Tenzer et al. , 2012cTenzer et al. , 2015)).Moreover, gravity signals associated with deeper masses below the crust (i.e.non-isostatic effects) must also be taken into account.
To do so, we use the method presented by Bagherbandi & Sjöberg (2012) to estimate the non-isostatic correction/effect (NIE) on gravity using the recent seismic crustal thickness in CRUST19 model by comparing the gravimetric and seismic data in the frequency domain.CRUST19 is a global crustal model based on a USGS database of crustal seismic studies.CRUST19 was also used to obtain the MD (D 0 ) and the NIE.

Validation of CRUST1.0 model versus CRUST2.0 model
As we will partly rely on CRUST1.0 model for our application, we start to compare it with CRUST2.0.First, it should be noted that the resolution of CRUST1.0 is 1 • × 1 • , while that for CRUST2.0 is 2 • × 2 • .CRUST2.0 is an updated version of the CRUST5.1 model (Mooney et al. 1998), it incorporates 360 key crustal types that contain the thickness, density and velocity of compressional (V p ) and shear waves (V s ) for seven layers (ice, water, soft sediments, hard sediments, upper, middle and lower crust).The V p values are based on field measurements, while V s and density are estimated by using empirical V p -V s and V p -density relationships, respectively.In 2013, CRUST2.0 was updated to CRUST1.0.Apart from the higher resolution CRUST1.0 is based on ETOPO1 for topography and bathymetry, sediment basins are taken from a 1 • × 1 • model (Laske & Masters 1997), while the crustal thickness is a compilation of active source experiments, receiver functions and already published Moho maps.
(1) The first step of the comparison is to compute the difference between the two models, M (1) i, j and M (2) i, j , for each grid cell (i, j)as where M (1) i, j and M (2) i, j are assumed to be the CRUST1.0 and CRUST2.0models.We then consider a statistical test, which shows how well M (1) i, j is statistically related to M (2) i, j .Therefore, we define a null hypothesis (H 0 ) and an alternative hypothesis (H a ) as follows: where η (1) i, j and η (2) i, j are the expected values of the models.The test parameter, which is approximately t-distributed, is given by where is the standard error of the mean difference between the models.
The standard errors in the CRUST1.0 and CRUST2.0models are defined empirically, implying 5 and 10 per cent of the calculated depth of the Moho as stated in Christensen & Mooney (1995).
For a given confidence level of 1 − α (α chosen as 0.05) and degree of freedom, df, the null hypothesis is met for where t α 2 is the critical value in the upper and lower tail of the t-distribution, and otherwise H a holds.
The MDs of the considered models are shown in Figs 1(a) and (b).
Generally, one can point out that the main differences between CRUST2.0 and CRUST1.0 are in the number of crustal types that decreases from 360 to 35, in the change of the upper mantle density model and in the introduction of mid oceanic ridges.In Fig. 1(c) one can observe that main differences between CRUST2.0 and CRUST1.0MDs in Ross ice shelf and Marie Byrd land are of the order of −15.9 km in mean value with a standard deviation of 4 km, partially due to the different resolutions and partially due to the different data set.Other differences between two models are located mainly in the areas of transition between continental and oceanic crust and in the presence of young oceanic crust (see also Sampietro et al. 2013).
Fig. 1(d) plots the coherence between CRUST1.0 and CRUST2.0MD, which follows from eq. (9d).The major advantage of using this plot is that it easily allows to detect the presence of anomalies in the crust (e.g.subduction, duplication and fragmentation of the crust).In the figure yellow areas indicate that the null hypothesis in eq. ( 9d) is rejected, implying that the two models are not consistent.This result may have two causes: either that the two estimated MD differ or that the estimated model accuracy is wrong, or both.With reference to Figs 1(a)-(d) one can see that CRUST1.0 agree to 4 km in RMS with CRUST2.0, in 88 per cent of the area.
(1) We also present the scatter plots between CRUST1.0 and CRUST2.0models to evaluate CRUST1.0 versus the other model.Scatter plots can usually be used to identify relationship between models.As can be seen, in order to figure out the relationship between pairs of the models, a line that best fits all the data (or regression line) has been performed (i.e.red middle line).The two additional lines (i.e.top and bottom lines) are drawn as ± twice of RMS differences between the models (i.e.fitted line ± 2 RMS differences) to assess the variation between the two models.Note the general linear correlation of the models in the plot indicating the overall agreement.One can also see outliers (i.e.plots outside the outer lines), which might be attributed to the improvements considered in CRUST1.0.
The validation of the global Moho models of CRUST1.0/CRUST2.0 is not an easy task due to the small amount of direct observations, (e.g. from seismic techniques) and to their limited accuracy, but nevertheless according to our validations, it is concluded that CRUST2.0 is a comparatively poor model for Antarctica (see also Abrehdary et al. 2017).
Table 1 shows some statistics of the free-air and refined Bouguer gravity disturbances and corrections, and more details on these disturbances are visualized in Figs 3(a)-(g).Fig. 3(a) depicts that the Antarctic free-air gravity disturbances differ mostly within ±80 mGal, with the largest values over the Antarctic Peninsula and large parts of East Antarctica, and small values mainly over the West Antarctic Rift System and the Transantarctic Mountains.
The resulting refined Bouguer disturbance vary between 5.8 and 424.5 mGal on the continent and between −585.9 and −21.3 mGal on the surrounding seas (Fig. 3g).As can be seen from Figs 3(a) and 3(g), these corrections significantly change the Bouguer gravity disturbance from the free-air disturbance over oceans due to the application of the bathymetric stripping gravity correction (Fig. 3c), and in continental Antarctica due to application of the ice density variation stripping gravity correction (Fig. 3d), and the change is less remarkable when applying the sediment density variation stripping gravity correction (Fig. 3e).Notable is also the large NIE in some ocean areas.The most pronounced effect of the NIE due to remaining GIA is naturally found in areas with the highest post-glacial rebound (cf.Section 5).

The variable Moho density contrast and its uncertainty
Table 2 summarizes some statistics of the MDC of HVMDC20 and its STE estimated by eq. ( 8), and details are shown in Figs 4(a) and (b).The MDC varies between 81 kg m −3 in some remote areas of the open sea to high values in the NE of the continent, reaching 579 kg m −3 in central Antarctica (in the Gamburtsev Mountain Range), and its STE is generally below 50 kg m −3 , varying between  8).From the figure one can observe that most of the STDs are less than 50 kg m −3 , but they reach to more significant values (up to 80 kg m −3 ) in some regions.

HVMDC20 assessment
Here we compare our results with those from the models KTH11, GEMMA12C, CRUST1.0 and KTH15C.Table 3 lists some properties of the models used in the comparison, and Table 4 shows the statistics of the comparisons of the MDCs with HVMDC20.Details of comparisons are shown in Figs 5(a)-(d).
Fig. 5(a) displays that HVMDC20 is not in good agreement with model KTH11, which could be explained by that neither the density variations related to the Earth crust, nor those related to the upper mantle were considered in KTH11, leading to an underestimation of the MDC (Abrehdary et al. 2017), which seems to be more realistic in the HVMDC20 model.
Fig . (5b) demonstrates that HVMDC20 and GEMMA12C are inconsistent in the ocean regions, while the agreement is acceptable on the continent.The discrepancies in the Antarctic seas could be attributed to the use of the generally inferior CRUST2.0 in the GEMMA12C and also to the differences in data resolution, and weighting of the different contributions (Reguzzoni et al. 2013).
Fig. 5(c) illustrates that the HVMDC20 agrees fairly well with CRUST1.0.The largest disagreements, of the order of 100 kg m −3 are found in the SE of the continent and in a small ocean area to the NW.These deviations may be attributed to the constraints used in  the CRUST1.0,which gives little information on Antarctic interior (Llubes et al. 2018).In Fig. 5(d), one can observe that although the KTH15C model shows a very good agreement with ours, however, some disagreements are found in some regions, which may be due to using erroneous prior information in the KTH15C and/or imperfect estimates in HVMDC20 in these areas.
In Fig. 6, the differences between HVMDC20 and MDCs from (a) KTH11, (b) GEMMA12C, (c) CRUST1.0 and (d) KTH15C model divided by the standard deviation of our HVMDC20 model are plotted, which can help us to localize specific regions, where the disagreements with KTH11, GEMMA12C, CRUST1.0 and KTH15C are bigger than ±1σ .As can be seen from Fig. 6(a), HVMDC20 agree rather poor with KTH11, which could be attributed on one hand to density variations related to the Earth crust and those related to the upper mantle were not considered in KTH11, which probably led to an underestimation of the MDC, that seems to be more realistic in the HVMDC20 model.Another problem was the use of the inferior CRUST2.0 model in KTH11 (but CRUST1.0 in HVMDC20), as the difference between the CRUST2.0 and CRUST1.0 is significant as shown in Figs 1(a)-(d).Fig. 6(b) shows that HVMDC20 and GEMMA12C are not in good consistency, which again can partly be explained by the utilizing the CRUST2.0model in GEMMA12C and also to the differences in data resolution.Figs 6(c) and (d) reveal that HVMDC20 is compatible with two models (CRUST1.0 and KTH15C), and in most areas the agreements are within ±1σ .However, in some regions they disagree significantly, which may be due to erroneous prior information in KTH15C and CRUST1.0uncertainties in these areas.

B R I E F OV E RV I E W O F H V M D C 2 0 I N A N TA RC T I C A
In order to better assess the MDC modelling by HVMDC20, we employ the thematic map plotted in Fig. 7, which shows elevations of East and West Antarctic ice caps, principal East Antarctic shield geology, modern glacial flow, and study areas used for sampling of sedimentary-and glacially transported proxies.
Fig. 4(a) shows that small MDCs, less than 150 kg m −3 , are found in ocean areas, which is obviously due to the thinning of the oceanic crust, particularly along the Pacific Antarctic mid-oceanic ridge.As can also be seen from Fig. 4(a), the MDCs change significantly between East and West Antarctica, with higher values in the East, particularly under the Gamburtsev Subglacial Mountains, confirming the presence of deep and compact orogenic roots.Comparing with Fig. 7, one can see that the MDCs are fairly notable in East Antarctica under Dronning Maud Land with two orogenic roots under Wohlthat Massif and the Kottas Mountains, that are separated by a relatively thin crust along Jutulstraumen Rift (see also Baranov et al. 2018).Large MDCs in East Antarctica, exceeding 500 kg m −3 , are accumulated throughout the central part, which is even extended under the Transantarctic Mountains.This is because, apart from small areas along the coast, East Antarctica is permanently covered by ice.The MDC in West Antarctica is also significant but less pronounced than that in the Eastern counterpart.Especially in SW Antarctica the MDC is typically 400-450 kg m −3 , which is attributed to the presence of the West Antarctic Rift system, extending over a 3000 × 750 km 2 , with large ice-covered areas from the Ross Sea to the base of the Antarctic Peninsula.The MDCs observed under the mountain ranges and areas covered by the largest continental ice sheet are also significant.

G L A C I A L I S O S TAT I C A D J U S T M E N T ( G I A ) OV E R A N TA RC T I C A
This section models the remaining glacial isostatic adjustment (GIA) effect on gravity to figure out how this affects MDC determination.GIA is the ongoing movement of land once loaded by ice-age glaciers.The process of GIA refers to isostatic deformation related to ice and water loading during a glacial cycle.Isostasy involves adjustment of the crust responding to the mass transport across its surface (Watts 2001, p.48), implying that the crust forms in response to the gravitational effect of dynamic processes due to GIA.Usually, the GIA effect is part of the NIEs, and the observed gravity disturbances are already corrected for it, but here we investigate specifically the GIA impact for Antarctica to figure out how much of the GIA signal can affect the MDC determination.To achieve this, we use the spherical harmonic spectral window 7-12 of the gravity field, which yields the largest correlation coefficient (∼0.60) with the GIA model W12a (Whitehouse et al. 2012).Applying this spectral window to the gravity disturbance and MDC (based on the VMM technique) is visualized in Table 5 and Fig. 8, respectively.
The remaining GIA is a non-isostatic gravity signal, which will falsely affect the MDC results if not considered.However, with reference to Table 2 and Fig. 8, one can see that the GIA effect on gravity (6.7 mGal in the STD) and on MDC (5.8 kg m −3 in the STD) versus the STD of the estimated MDCs (123 kg m −3 ) are not significant.Consequently, we conclude that the remaining GIA has no significant effect on the estimated MDC over Antarctica.

C O N C L U S I O N S
The VMM technique has been applied successfully in the new MDC model HVMDC20 with a resolution of 1 • × 1 • in Antarctica and surrounding seas using the recent gravity field model XGM2019e and Earth2014 topographic/bathymetric information along with CRUST1.0 and CRUST19.The model is also attached with a map of its estimated STE, (but possible significant systematic errors are missing).The STE of the oceanic MDC is typically 10 kg m −3 or less, while on the continent it reaches more than 80 kg m −3 at some places.
To validate the results, we have compared HVMDC20 with the similar models KTH11, GEMMA12C, CRUST1.0 and KTH15C, showing that the last two models agree well with HVMDC20.In addition, we have studied the signatures of geological unit structures in HVMDC20, showing how the main geological and ice sheet structures attribute in the MDC.Finally, we have specifically investigated the remaining GIA effect on gravity to figure out how it affects MDC recovery, yielding a correlation of the order of ∼0.6 with the W12a GIA model and contributing within ±14 kg m −3 to the MDC.

A C K N O W L E D G E M E N T S
This study was supported by project no.187/18 of the Swedish National Space Agency (SNSA).Prof. John Goodge from the University of Minnesota is acknowledged for providing the geology map of Antarctica.

Figure 1 .
Figure 1.(a) MD derived by the CRUST1.0,(b) MD obtained from CRUST2.0,(c) differences between CRUST1.0 and CRUST2.0MD and (d) coherence (or Fitness) between CRUST1.0 and CRUST2.0MD model.Yellow pixels represent the grid cells where the hypothesis presented in eq.(9d) is rejected, that is, the two models are not consistent.RMS is root mean square of the difference between the two models.

Figure 3 .
Figure 3. (a) The free-air gravity disturbances produced by the XGM2019e; (b) the topographic gravity correction; (c) the bathymetric gravity correction; (d) the ice density variation gravity correction; (e) the sediment density variation gravity correction; (f) the NIEs on gravity disturbance; and (g) the refined Bouguer gravity disturbances (in mGal).

Fig. 2
Fig.2depicts scatter plots of the global MD models CRUST1.0 versus CRUST2.0.As can be seen, in order to figure out the relationship between pairs of the models, a line that best fits all the data (or regression line) has been performed (i.e.red middle line).The two additional lines (i.e.top and bottom lines) are drawn as ± twice of RMS differences between the models (i.e.fitted line ± 2 RMS differences) to assess the variation between the two models.Note the general linear correlation of the models in the plot indicating the overall agreement.One can also see outliers (i.e.plots outside the outer lines), which might be attributed to the improvements considered in CRUST1.0.The validation of the global Moho models of CRUST1.0/CRUST2.0 is not an easy task due to the small amount of direct observations, (e.g. from seismic techniques) and to their limited accuracy, but nevertheless according to our validations, it is concluded that CRUST2.0 is a comparatively poor model for Antarctica (see alsoAbrehdary et al. 2017).Table1shows some statistics of the free-air and refined Bouguer gravity disturbances and corrections, and more details on these disturbances are visualized in Figs 3(a)-(g).Fig.3(a) depicts that the Antarctic free-air gravity disturbances differ mostly within ±80 mGal, with the largest values over the Antarctic Peninsula and large parts of East Antarctica, and small values mainly over the West Antarctic Rift System and the Transantarctic Mountains.

Figure 7 .
Figure 7. Map of Antarctica showing elevations of East and West Antarctic ice caps (Lythe et al. 2001), principal East Antarctic shield geology (after Fitzsimons 2000), modern glacial flow (after Barker et al. 1998) and study areas used for sampling of sedimentary and glacially transported proxies.Warmer colours indicate higher ice surface elevations (Goodge 2007).

Table 1 .
Statistics of global estimates of the gravity disturbances, stripping gravity corrections and NIEs.STD is the standard deviation of the estimated quantities in Antarctica.δg FA XGM2019e is the free-air gravity disturbance provided by the XGM2019e.δg Topo , δg Bath , δg Ice and δg Sedi are the topographic/bathymetric, ice and sediment stripping gravity corrections derived from ESCM180.δg NIES is the non-isostatic effect estimated by the CRUST19, and δg RFB is the refined Bouguer gravity disturbance after applying the stripping gravity corrections and NIEs.Unit: mGal.

Table 2 .
Statistics of the MDC and its uncertainty estimated from HVMDC20 model.STD is the standard deviation of the estimated quantities beneath Antarctic.ρ HVMDC20 and σ ρ HVMDC20 are the MDC and its uncertainty estimated from HVMDC20 model.

Table 3 .
Source of MDC models used in this study.

Table 4 .
Statistics of the differences between the HVMDC20 and other global models.STD is the standard deviation.RMS is root mean square.KTH11, GEMMA12C and KTH15C are the MDCs used to compare with those from HVMDC20.

Table 5 .
Statistics of the GIA gravity and its effects on the MDC over Antarctica.STD is the standard deviation of the estimated quantities over the ocean blocks.δg GIA and ρ GIA are the GIA effects on gravity and MDC, respectively.