## 1. Introduction

Since the Glacial Isostatic Adjustment (GIA) caused by the melting of past ice sheets is still contributing to present-day regional and global sea-level variations, understanding its spatio-temporal variability is important to interpret the effects of present climate change. However, because GIA is also affecting the shape of the Earth and its gravity field, it has a clearly established role in the interpretation of a wide range of geophysical data, obtained by means of ground-based or space-borne geodetic observations. The relevance of GIA modeling in various fields of the Geosciences and its increasing role in the framework of global change constitutes the motivation of this study, mainly aimed at reviewing systematically the imprints of GIA and to describe their geometry in connection with the Sea Level Equation (SLE).

To introduce GIA, it is convenient to define a

reference state in which the solid Earth, the ice sheets and the oceans are in an equilibrium configuration, sketched in

Figure 1a, and to compare it to a subsequent perturbed state. This approach was originally proposed by Farrell and Clark [

1], hereafter referred to as FC76, in their seminal work where the SLE was introduced first. The reference configuration can be chosen arbitrarily, but for our discussion it is convenient to refer to the Last Glacial Maximum (LGM, ∼21,000 years ago). The load acting on the Earth’s surface in the reference state (i.e., the mass per unit area) is

${L}_{0}\left(\omega \right)$ and

${I}_{0}\left(\omega \right)$ is the ice thickness, with

$\omega =(\theta ,\lambda )$ where

$\theta $ is colatitude and

$\lambda $ is longitude. The SLE has can be employed to predict how sea level shall change at an arbitrary location

$\omega $, when the configuration of the system portrayed in

Figure 1a evolves in a

new state shown in

Figure 1b at time

$t\ge {t}_{0}$, in which the surface load and the ice thickness are

$L(\omega ,t)$ and

$I(\omega ,t)$, respectively. Despite the global variations observed in the new state, (i) the mass of the system (ice+oceans+solid Earth) must be conserved, and (ii) the new sea surface must remain an equipotential; ultimately, these are the two fundamental principles that the SLE makes manifest.

The interactions responsible for the changes observed in the new state are qualitatively sketched in the diagram of

Figure 2, freely modified from Clark et al. [

2]. Since the interactions are operating simultaneously and at all spatial scales, their contributions cannot be easily disentangled, which makes the interpretation of the GIA effects on sea level and on various geodetic quantities particularly challenging. In the top part, the figure is showing the three fundamental elements of the SLE, i.e., the ice sheets, the solid Earth, and the oceans [

3]. As indicated by the arrows, these elements are interacting by two mechanisms: (i) surface loading and (ii) mutual gravitational attraction.

The waxing and waning ice sheets, by changing the local ice masses, exert a load at the surface of the solid Earth (

ice loading, related to glacio-isostasy), but the mass variation of the oceans is also loading the Earth, acting on the seafloor (

water loading, associated to hydro-isostasy). These two non-uniform loads are tightly interconnected, since the mass conservation of the system (water + ice) imposes that, on average, the load variation vanishes across the Earth’s surface. Due to the mantle imperfect elasticity, the past loads also induce delayed and still persistent effects that are manifest as a global state of isostatic disequilibrium. Furthermore, the equipotential surfaces of the Earth’s gravity field are twisted by the mass redistributed over the solid Earth and in the oceans, causing variations of the geoid. The three elements that enter into the SLE are all affected by gravitational attraction. In particular, the sea surface is warped by the attraction of the continental ice sheets, but at the same time the geoid variations caused by the solid Earth deformation modify the shape of the oceans. The bottom part of

Figure 2 considers further interactions driven by the Earth’s irregular rotation. Inertia perturbations, associated to long wavelength deformations and sea-level variations of harmonic degree

$l=2$, drive excursions of the rotation axis in order to conserve the Earth’s angular momentum [

4]. The consequent variation of the centrifugal potential alters, in turn, both the solid Earth and the sea surface, a process that is known as

rotational feedback on sea level, see Peltier [

5].

The

inextricably related interactions first acknowledged by Clark et al. [

2] and illustrated in

Figure 2 are responsible for the regional imprints of GIA. As first noted by Woodward [

6] and later discussed by Daly [

7], Walcott [

8] and Farrell and Clark [

1], the sea-level variations associated with glacial isostasy depart significantly from the spatially uniform pattern that we would observe for a rigid, non-gravitating and non-rotating Earth (i.e., ignoring the interactions). Often, in the geological literature the spatially uniform sea-level change is referred to as

eustatic, a word attributed to Suess [

9]; eustatic variations only depend on the history of the past grounded ice volume [

10]. Presently, the term

barystatic is preferred [

11]. The interactions are responsible for a global pattern of relative sea level (RSL) variations during the melting of the late-Pleistocene ice sheets, which Clark et al. [

2] have characterized by defining six

RSL zones, labelled from

I to

$VI$ (see their Figure 5); within each zone, the sea-level signatures are similar to one another. The RSL zones encompass the glaciated areas (zone

I), the region of the collapsing fore-bulge (

$II$), the time-dependent emergence (

$III$) and the oceanic submergence zone (

$IV$), the oceanic emergence region (

V), and the continental shorelines (

$VI$). Subsequently, Mitrovica and Milne [

12] have studied the nature of the RSL zones in connection with the various terms of the SLE, describing the physical mechanisms responsible for their establishment and unveiling the processes of

continental levering and

ocean siphoning. Following the above studies, the spatial variability in sea level associated with GIA has been widely investigated with the aim of reconstructing the history of deglaciation since the LGM (see e.g., [

13,

14,

15]). On a more limited spatial scale, the concept of RSL zone has also been useful to interpret the Holocene sea-level variations across the Mediterranean Sea [

16,

17].

The study of paleo-shorelines has allowed to define the broad features of the pattern of RSL zones since the LGM (see, e.g., Lambeck and Chappell [

18]). However, the present-day trends of sea level detected at tide gauges or by satellite altimetry should be certainly also affected by contemporary variations in the state of the cryosphere driven by global warming. In this context, the question has not been addressed until the work of Plag and Jüettner [

19], who have first coined the term of

fingerprint (function). Quoting their same words,

…The elastic response of the Earth to present-day changes in the cryosphere can be expected to produce a similar fingerprint, which should be present in the tide gauge data. Based on these fingerprints, tide gauge trends, in principle, can be inverted for ice load changes [

19]. However, after having analyzed the relative sea-level trend for some long tide gauge time series, Douglas [

20] concluded that

unambiguous evidence for fingerprints of glacial melting was not found, most likely due to the presence of other signals present in sea-level records that cannot easily be distinguished. Recently, Spada and Galassi [

21] have quantitatively compared the harmonic power spectrum of contemporary sea-level change to that of GIA, including the contribution due to the disintegration of the past ice sheets and that associated to present deglaciation. They have shown that the

power of GIA from past ice melting is comparatively modest at all harmonic degrees, with the possible exception of harmonic degree

$l=2$, and it cannot emerge from the steric component that dominates current sea-level rise [

22]. Notwithstanding the difficulty of visualization, the concept of sea-level fingerprint has undoubtedly gained an important role in the interpretation of the trends of contemporary [

23,

24,

25,

26,

27] and future sea-level rise [

28,

29,

30].

In this work, we aim at exploring and reviewing the properties and the symmetries of the GIA fingerprints presently associated with the melting of past ice sheets, as well as the intercorrelations among them. Much of what we present in this paper can be also applied to the fingerprints of present ice melting, which obey the SLE as well; these have been discussed in various places, see e.g., [

31] and references therein. Although we are aware that uncertainties on the Earth’s viscosity profile and the chronology of deglaciation affect significantly the pattern and the amplitude of the GIA fingerprints [

32], here for simplicity we shall only consider a specific GIA model, leaving an error analysis to future work, along the lines of Melini and Spada [

32]. The paper is organized as follows. In

Section 2 we review the theory behind the SLE. In

Section 3 we briefly present the GIA model used and the numerical approach adopted. In

Section 4 we illustrate some of the properties of the present-day GIA fingerprints associated with the melting of the past ice sheets, which in

Section 5 are exploited to interpret the global uplift pattern of continents currently detected by GPS data. Our conclusions are drawn in

Section 6.

## 2. Theory

Here we briefly introduce the essentials of the SLE theory, necessary to illustrate the geometry of the GIA fingerprints in

Section 4 below. The reader is referred to Spada and Melini [

33] (hereinafter SM19) and to its supplement for a more detailed and self-contained presentation. The paper SM19 and its supplement are presently (August 2019) submitted to Geoscientific Model Development (GMD), the interactive open-access journal of the European Geosciences Union at

https://www.geosci-model-dev-discuss.net/gmd-2019-183/. The open-source program SELEN

${}^{4}$ (SELEN version 4.0) can be obtained from

https://zenodo.org/record/3377404. We note that the SLE theory does not account for tectonic deformations nor for variations in the temperature or salinity of the ocean water, which we do not consider in our analysis.

In the reference state considered in

Figure 1a,

sea level is defined by the difference

where

$\omega =(\theta ,\lambda )$ are the coordinates of a given point on the Earth’s surface, while

${r}_{0}^{ss}\left(\omega \right)$ and

${r}_{0}^{se}\left(\omega \right)$ are the radii of the (equipotential) sea surface and of the solid Earth in a geocentric reference frame with origin in the whole-Earth center of mass, respectively. As shown in

Figure 1a,

${B}_{0}$ would be directly measured by a stick meter, i.e., a

tide gauge, placed at

$\omega $. Assuming that the horizontal displacement of the stick-meter has been negligible in comparison to vertical displacement, in the new state sea level is

where

${r}^{ss}(\omega ,t)$ and

${r}^{se}(\omega ,t)$ denote the new radius of the equipotential sea surface and of the solid surface of the Earth, respectively. Note that

topography is related to sea level through

Combining (

2) with (

1),

relative sea-level change
can be also expressed as

where

is the

sea surface variation, or

absolute sea-level change, and

is the

vertical displacement of the Earth’s surface. Equation (

5) represents the most basic form of the SLE. We note that, being defined as a double difference, relative sea-level change

$\mathcal{S}(\omega ,t)$ is not dependent upon the choice of the origin of the reference frame, i.e., it is an absolute quantity. Quantities

$\mathcal{N}(\omega ,t)$ and

$\mathcal{U}(\omega ,t)$, however, depend on the choice of the origin.

The sea surface variation

$\mathcal{N}(\omega ,t)$ is tightly associated to the variation of the geoid height. However, as remarked by FC76,

$\mathcal{N}(\omega ,t)$ is not

the variation of the geoid height: i.e.,

on a rigid Earth model, there is no distinction between changes in geoid radius and changes in sea level, but it is important to realize the difference between these quantities for deformable Earth models [

1]. A further problem arises from the fact that, in the new state, the volume of the oceans is varied to compensate the mass lost or gained by the continental ice sheets. Indeed, as pointed out by Tamisiea [

34], some confusion arose recently about the definition of

$\mathcal{N}(\omega ,t)$, which sometimes is still used as a synonymous of geoid height variation; the confusion is attributed to often inconsistent terminology between various disciplines. FC76 have shown that the sea surface height variation is

where

is the

variation of the geoid radius relative to the reference state,

$\mathsf{\Phi}(\omega ,t)$ is the variation of the total gravity potential of the Earth system, taking both surface loading and rotational contributions into account,

g is the reference surface gravity acceleration and

c is a yet undetermined spatially invariant term, notorious within the GIA community as the

FC76 c-constant. In the following, Equation (

8) shall be referred to as

FC76 formula. Thus, using Equation (

8) in (

5), the SLE reads

where we have defined the

sea-level response function as the difference

It is now convenient to average both sides of Equation (

10) over the oceans, where the ocean-average of any function

$F(\omega ,t)$ is defined, at time

t, as

where

${\int}_{o}$ denotes the integral over the time-dependent surface of the oceans,

${A}^{o}$ is their area at time

t,

$dA={a}^{2}sin\theta d\theta d\lambda $ is the element of area over the surface of the sphere, and

a the average Earth’s radius. We recall that the surface of the oceans is the region where

$O=1$, where

O is the

ocean function (OF) defined as

where

${\rho}^{i}$ and

${\rho}^{w}$ are the densities of ice and water, respectively. For

$O=1$, the ocean is ice-free, or there is floating ice; for

$O=0$, the ice is grounded either below or above sea level, or the land is ice-free. Using a

continent function defined as

$C(\omega ,t)=1-O$ is sometimes useful. Since

$<c{>}^{o}\equiv c$, solving Equation (

10) with respect to the FC76 constant gives

where we have defined

${\mathcal{S}}^{ave}\equiv <\mathcal{S}{>}^{o}$. Hence, using Equation (

14) into (

10), the SLE is further transformed into

The response function

$\mathcal{R}(\omega ,t)$ embodies all the interactions qualitatively described in

Figure 2; following SM19, we split it into a contribution due to surface loads

and gravitational attraction (labeled by

sur) and one due to rotational effects (

rot), with

where

${\mathcal{R}}^{sur}(\omega ,t)={\mathcal{G}}^{sur}-{\mathcal{U}}^{sur}$ and

${\mathcal{R}}^{rot}(\omega ,t)={\mathcal{G}}^{rot}-{\mathcal{U}}^{rot}$. According to Farrell [

35],

${\mathcal{R}}^{sur}$ is given by a 3-D spatio-temporal convolution that involves the

surface Green’s function for sea level ${\Gamma}^{s}$ and the

surface load variation $\mathcal{L}=L-{L}_{0}$, where

L and

${L}_{0}$ denote the surface load in an arbitrary state and in the reference state, respectively, and

while following Milne and Mitrovica [

36],

${\mathcal{R}}^{rot}$ can expressed as a 1-D time convolution between the

rotation Green’s function for sea level ${{\rm Y}}_{l}^{s}$ and the

centrifugal potential variation, with

where (

$l,m$) are the spherical harmonic degree and order, respectively (with

$l=0,1,2,\dots ,{l}_{max}$ and

$\left|m\right|\le l$). Han and Wahr [

37] and Milne and Mitrovica [

36], however, have shown that

$\mathsf{\Lambda}(\gamma ,t)$ is essentially a spherical harmonic function of degree and order

$(l,m)=(2,\pm 1)$. The Green’s functions

${\mathsf{\Gamma}}^{s}$ and

${\mathsf{{\rm Y}}}_{l}^{s}$ are expressed by particular combinations of

loading Love numbers and

tidal Love numbers, respectively. It is important to note that the harmonic coefficients of

${\mathcal{R}}^{sur}(\omega ,t)$, i.e.,

${\mathcal{R}}_{lm}^{sur}\left(t\right)$, depend linearly from those of the surface load variation

${\mathcal{L}}_{lm}\left(t\right)$ (see supplement of SM19 for details).

An explicit expression for

${\mathcal{S}}^{ave}$ in Equation (

15) is obtained applying the mass conservation principle that according to SM19 can be stated in various equivalent ways. Here it is convenient to use the form

where the average over the whole Earth’s surface is defined, in analogy with Equation (

12), as

$<\cdots {>}^{e}\left(t\right)\equiv (1/{A}^{e}){\int}_{e}(\cdots )\phantom{\rule{3.33333pt}{0ex}}dA$. We refer to mass conserving loads that obey Equation (

19) as

physically plausible loads. As shown in SM19, condition (

19) is equivalent to

where

${\mathcal{L}}_{00}\left(t\right)$ is the spherical harmonic component of the surface load for degree and order

$(l,m)=(0,0)$. Using the result

(SM19) and some algebra, from the constraint of mass conservation we obtain

where

${\mathcal{S}}^{equ}$ (equivalent sea-level change) is defined as

with

$\mu \left(t\right)={\rho}^{i}{\int}_{e}\left(IC-{I}_{0}{C}_{0}\right)dA$ denoting the time variation of the grounded ice mass, and term

is associated with ocean function variations, where

${T}_{0}$ and

${O}_{0}$ are the initial topography and the initial OF, respectively. We note that in the fixed-shorelines approximation of FC76, the OF is constant, with

$O={O}_{0}={O}^{p}$ where

${O}^{p}$ is the present OF. Hence, in this approximation

${\mathcal{S}}^{ofu}=0$, and

${\mathcal{S}}^{equ}$ is equivalent to what in the geological literature is often called

eustatic [

9] sea-level change

where

${A}^{op}={\int}_{e}{O}^{p}\phantom{\rule{0.166667em}{0ex}}dA$ is the present-day area of the oceans.

The SLE (

15), complemented by Equations (

16)–(

18) and (

22) constitutes a 3-D non-linear integral equation in the unknown

$\mathcal{S}(\omega ,t)$, somewhat similar to a 1-D non-homogeneous Fredholm equation of the second kind (see e.g., [

38]). Assuming fixed shorelines, as in FC76, would reduce the SLE to a linear equation [

31]. The integral, or implicit, nature of the SLE becomes apparent when it is recognized that the response function

$\mathcal{R}$ functionally depends, through

$\mathcal{G}$ and

$\mathcal{U}$, upon

$\mathcal{S}$ itself (see SM19). In modern approaches to GIA, the SLE is solved recursively in the spectral domain, adopting the

pseudo-spectral method [

39,

40].

In the general case given by Equation (

15), no analytical solutions exist for the SLE. However, a closed-form solution can be found in the eustatic approximation, expressed by (

25), valid in the very special case of a rigid Earth in which the gravitational attraction between the three components of the SLE is neglected (see

Figure 2). Another analytical solution is found assuming a rigid Earth and uniform oceans but allowing for the gravitational interaction between the ice sheets and the oceans, i.e., neglecting the self-attraction of oceans. This solution, often referred to as

Woodward solution [

6], has been discussed in detail by e.g., Spada [

31]. Although oversimplified, it is historically important and it has the merit of demostrating the important role of gravitational attraction in shaping the sea surface, with a sea-level change departing from the spatially uniform eustatic solution both nearby the melting ice sheets and in their far field.

## 3. Methods

Spada and Melini [

33] have recently released a general open-source Fortran program called SELEN

${}^{4}$ (SELEN version 4.0) that solves the SLE in its full form; this shall be employed in next sections to study the geometry of the GIA fingerprints associated with the melting of past ice sheets. SELEN

${}^{4}$ is the current stage of the evolution of program SELEN which was originally published in 2007 by Spada and Stocchi [

41] based upon the theory detailed in Spada and Stocchi [

42].

SELEN

${}^{4}$ implements the pseudo-spectral method of Mitrovica and Peltier [

39] and Mitrovica and Milne [

40]. In SELEN

${}^{4}$, all the variables have a piecewise constant time evolution. In space, the discretization is performed adopting the equal-area icosahedron-based spherical geodesic grid designed by Tegmark [

43], whose density is controlled by the resolution parameter

R. In our computations, we have set

$R=44$, corresponding to

$P=40R(R-1)+12=$ 75,692 pixels over the sphere, each having a radius of ∼46 km. In this way, the number of cells is comparable to that of a traditional

${1}^{\circ}\times {1}^{\circ}$ spherical grid, i.e., 64,800. The spherical harmonic expansions required in the framework of the pseudo-spectral approach are truncated at degree

${l}_{max}=128$ and the coefficients are evaluated taking advantage of the quadrature rule for the Tegmark grid [

43]. According to SM19, the chosen combination (

$R,{l}_{max}$) ensures a sufficient precision without being computationally too demanding.

In SELEN

${}^{4}$, we have implemented the GIA model ICE-5G of Peltier [

44]. The ice thickness has been discretised on the Tegmark grid and reduced, at a given pixel, to a uniform sequence of identical time steps with a length of 500 years. The LGM is at 21 ka, and prior to that isostatic equilibrium is assumed. Since the LGM, ICE-5G releases a total equivalent sea level

where we have assumed

${\rho}^{i}=931.0$ and

${\rho}^{w}=1000.0$ kg m

${}^{-3}$, and

$\Delta {V}^{i}$ is the ice volume variation since the LGM. We combine the ICE-5G deglaciation history with a three-layer volume-averaged version of the VM2 multi-stratified rheological profile [

44]. The Maxwell viscosities are

$\eta =2.7$,

$0.5$ and

$0.5$ in units of

${10}^{21}$ Pa·s in the lower mantle, transition zone and shallow upper mantle, respectively. We assume a fluid inviscid core, a 90 km thick elastic lithosphere and an incompressible rheology. A PREM-averaged [

45] density and rigidity profile has been adopted, using a 9-layer radial structure (see

Table 1). Loading and tidal Love numbers have been computed using program TABOO [

46] in a multi-precision environment [

47], and expressed in a geocentric reference frame with origin in the center of mass of the whole Earth, including the solid and the fluid portions. To facilitate reproducibility, some values of the Love numbers are listed in

Table 2.

The SLE has been solved iteratively [

48] adopting three “external” iterations to progressively refine the OF and the paleo-topography and, for each of them, performing three “internal” iterations to solve for

$\mathcal{S}(\omega ,t)$, for a given approximation of topography. According to SM19 and to independent results by Milne and Mitrovica [

36], these choices ensure sufficiently precise results. The present-day relief, obtained by a pixelization of the ice-free version of model ETOPO1 [

49,

50], has been imposed as a

final condition. The present-day ice distribution is given by the last time step of ICE-5G. Finally, to model the effects of polar motion on sea-level change, we have employed the

revised rotation theory by Mitrovica et al. [

51] and Mitrovica and Wahr [

52]. The revised theory accounts, in the polar motion equations [

4], for the long-term viscous behavior of the lithosphere and for the effects of internal dynamics on the Earth flattening. Some runs, however, have been performed adopting the

traditional rotation theory (see e.g., Spada et al. [

46] and references therein), which assumes an elastic lithosphere in the polar motion equations and does not take internal dynamics into account. In some other runs, we are totally neglecting the effects of Earth rotation.

## 5. Observing the Global GIA Fingerprint by Vertical GPS Rates

As an example of application of the fingerprints properties illustrated above, we mention the problem of directly using geodetic observations to quantify the present global pattern of GIA. Recently, using a large global compilation of geodetic GPS rates in conjunction with a Bayesian inference method, Husson et al. [

72] have reconstructed and visualized the long-wavelength signature of GIA on the rate of present day vertical crustal uplift. In principle, once the contributions from short-wavelength tectonic phenomena have been filtered out, the geodetically observed rates across the continents should match those predicted by current GIA models, at least in their global traits and in their spatial averages. One possible way to verify this consistency is to consider the average over the continents of the geodetically determined rate of vertical uplift

$<\dot{\mathcal{U}}{>}^{c}\left(t\right)$, where

$<\cdots {>}^{c}\left(t\right)\equiv (1/{A}^{c}){\int}_{c}(\cdots )\phantom{\rule{3.33333pt}{0ex}}dA$,

${A}^{c}\left(t\right)={A}^{e}-{A}^{o}$ being the area of the continents. In Husson et al. [

72], the scalar field

$\dot{\mathcal{U}}$ has been estimated from GPS vertical rates by a self-adaptive trans-dimensional regression that exploits the properties of the Voronoi tesselation. The pattern of GIA inferred by regression has been found to broadly resemble the one that we would expect by a model like ICE-5G (VM2), provided that components with wavelengths <2500 km are removed.

Of course, the average

$<\dot{\mathcal{U}}{>}^{c}$ could be evaluated numerically using the results of

Figure 4 and compared to the value obtained from the pattern of the GPS rates. However, through the SLE, it is straightforward and possibly more meaningful to express

$<\dot{\mathcal{U}}{>}^{c}$ in terms of the ocean-averaged fingerprints

$<\dot{\mathcal{S}}{>}^{o}$ and

$<\dot{\mathcal{N}}{>}^{o}$ that we have already discussed above in view of their particular significance. Here we largely follow Husson et al. [

72] and, since we simply aim at illustrating the method, we do not consider the modeling uncertainties on the GIA fingerprints. On one hand, taking advantage of Equation (

29), we have

which by the additivity of the surface average gives

or, equivalently,

where we have used the definitions of continent and ocean average; hence

On the other hand, ocean-averaging both sides of the SLE in the form (

5) gives

which used into (

39) yields the average of

$\dot{\mathcal{U}}$ across the continents

which is only expressed in terms of ocean-averaged fingerprints.

The exercise above shows that the average vertical uplift across the continents determined by GPS could be estimated,

in principle, by ocean-averaging the tide gauge trends and subtracting the ocean averaged altimetry-derived rate of absolute sea-level change. This would hold true regardless the GIA model employed. By approximating

${A}^{o}\approx (7/10){A}^{e}$ and

${A}^{c}\approx (3/10){A}^{e}$, so that

${A}^{o}/{A}^{c}\approx 7/3$, and using the ocean averages for ICE-5G (VM2) given by Equations (

27) and (

33), we obtain the not small value

where we note that since in Equation (

41) Husson et al. [

72] have used the crude fixed-shorelines approximation that implies

$<\dot{\mathcal{S}}{>}^{o}=0$ they have obtained slightly different values for

$<\dot{\mathcal{U}}{>}^{c}$. Nevertheless, the value of

$<\dot{\mathcal{U}}{>}^{c}$ matches very well the rate effectively reconstructed by Husson et al. [

72]

$(0.64\phantom{\rule{3.33333pt}{0ex}}{\mathrm{mm}\mathrm{year}}^{-1})$. This suggests that the trans-dimensional regression method has been effective in isolating the fingerprint of GIA. As pointed by Husson et al. [

72], the effects of current melting of glaciers and ice caps in response to global warming would not alter substantially our estimate in (

42).

## 6. Conclusions

In this work we have reviewed some aspects of GIA, i.e., the response of the Earth to the disequilibrium caused by the melting of the late-Pleistocene ice sheets. Arguments based upon the physical properties of the SLE have been corroborated by results obtained from up-to-date numerical tools in GIA modeling. Among the processes that concur to present sea-level rise, the special role of GIA has been recognized long ago; in fact, only GIA is affected by the rheology of the Earth and, at the same time, it affects significantly the gravity field and the rotational state of the planet. Although, according to GIA models, the deglaciation of the late-Pleistocene ice sheets came to an end thousands of years ago, at present the effects of GIA are still significant and they influence a number of directly observable geophysical and geodetic quantities. Since GIA evolves slowly, its contribution to the instrumental observations will persist also during next centuries although it shall gradually fade away. Model predictions show that the computed patterns or fingerprints of GIA are characterized by an outstanding complexity. In our roundup of the general properties of the GIA fingerprints, we have considered both the geometrical and the physical aspects of such complexity, emphasizing their spatial symmetries and regional character, which we have interpreted qualitatively and quantitatively with the aid of the SLE.

The study of the relative sea level fingerprint has revealed that at present the role of GIA is not that of causing an effective, mean global change. Rather, it causes essentially local regional effects which strongly contaminate the tide gauges records but that almost wash out when averaged over the present-day oceans, leaving a small contribution reflecting minor variations in the area of the sea floor and possibly current variations of ice thickness, when these are accounted for by the GIA model. The coastal regions are certainly those being most affected by the regional variability of GIA. The pattern of the GIA-induced vertical uplift is extremely variegated but it globally averages out to zero as an effect of mass conservation. Although it is largely anti-correlated to that of relative sea level, it shows more clearly the mark of the rotational effects of GIA, with a symmetry dominated by a very long-wavelength harmonic pattern. By a specific example from the recent literature, we have shown that the basic traits of the GIA fingerprint of vertical displacement can be visualized using data from a large global compilation of geodetic GPS rates. The symmetry imposed by the polar drift of the rotation axis is even more enhanced when one considers the fingerprints of the geoid height variation and of the absolute sea-level change, which only differ by a spatially invariant term. These two last signatures of GIA have presently a particular role in physical geodesy, since they are commonly employed to purge the trends of the Stokes coefficients of the gravity field and the sea-level altimetric records from the GIA effects.

In this study, we have employed only one of the ICE-

X models of WR Peltier and collaborators [

44], leaving a systematic exploration to a further study. Unquestionably, the shapes of the GIA fingerprints are significantly affected by the choice of the deglaciation chronology and of the rheological profile. An idea of this sensitivity can be obtained comparing directly the results obtained here and based on ICE-5G (VM2) with those of SM19, who have adopted a realization of ICE-6G (VM5a) [

57]. In [

32], it is possible to visualize the GIA fingerprints for old model ICE-3G (VM1) of Tushingham and Peltier [

54], which differs from the most recent ICE-

X models for a significantly larger meltwater release from Antarctica. For a discussion about the relationship between the fingerprints shapes and the GIA models employed, see in particular Spada and Galassi [

73], who have emphasized how the choice of the model affects the GIA corrections at tide gauges and, consequently, the estimate of secular sea-level rise. Of course, GIA models that are independent from the ICE-

X models exist and have a central role in the literature, like those progressively developed at the National Australian University by Kurt Lambeck and colleagues (see Nakada and Lambeck [

74], Lambeck et al. [

75] and subsequent contributions). Some GIA fingerprints for one of these models are shown in [

32].

Recent reviews [

31,

76] have indicated that the evolution of the GIA models has been considerable during last decades, because of the increased availability of proxy data constraining the history of sea level in the last thousand years [

31,

34]. Such evolution has also motivated efforts aimed at extracting geophysical information from ensembles (or more often mini-ensembles) of GIA models, as done by e.g., [

32,

77,

78,

79,

80,

81,

82]. The evolution of GIA models shall certainly continue in the future, in order to account for more realistic (possibly three-dimensional) descriptions of the Earth’s rheology, to include new details of the history of the ice sheets and their distribution, to relax some simplifying assumptions in the theory behind the SLE, to further fine-tune the rotation theory, and to add new elements or new branches to the interactions diagram of

Figure 1. Thus, although their general properties associated to the principle of mass conservation shall not change, the shape of the GIA fingerprints is certainly not given once and for all.