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**Geoscientific Model Development**
An interactive open-access journal of the European Geosciences Union

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**Model description paper**
04 Dec 2019

**Model description paper** | 04 Dec 2019

SELEN^{4} (SELEN version 4.0): a Fortran program for solving the gravitationally and topographically self-consistent sea-level equation in glacial isostatic adjustment modeling

^{1}Dipartimento di Scienze Pure e Applicate (DiSPeA), Università di Urbino “Carlo Bo”, Urbino, Italy^{2}Istituto Nazionale di Geofisica e Vulcanologia, Via di Vigna Murata 605, Rome, Italy

^{1}Dipartimento di Scienze Pure e Applicate (DiSPeA), Università di Urbino “Carlo Bo”, Urbino, Italy^{2}Istituto Nazionale di Geofisica e Vulcanologia, Via di Vigna Murata 605, Rome, Italy

**Correspondence**: Giorgio Spada (giorgio.spada@gmail.com)

**Correspondence**: Giorgio Spada (giorgio.spada@gmail.com)

Abstract

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We present SELEN^{4} (SealEveL EquatioN solver), an open-source program written in
Fortran 90 that simulates the glacial isostatic adjustment (GIA) process in response
to the melting of the Late Pleistocene ice sheets. Using a pseudo-spectral approach complemented
by a spatial discretization on an icosahedron-based spherical geodesic grid, SELEN^{4} solves a
generalized sea-level equation (SLE) for a spherically symmetric Earth with linear viscoelastic
rheology, taking the migration of the shorelines and the rotational feedback on sea level into
account. The approach is gravitationally and topographically self-consistent, since it considers
the gravitational interactions between the solid Earth, the cryosphere, and the oceans, and
it accounts for the evolution of the Earth's topography in response to changes in sea level.
The SELEN^{4} program can be employed to study a broad range of geophysical effects of GIA,
including past relative sea-level variations induced by the melting of the Late Pleistocene ice
sheets, the time evolution of paleogeography and of the ocean function since the Last Glacial
Maximum, the history of the Earth's rotational variations, present-day geodetic signals observed
by Global Navigation Satellite Systems, and gravity field variations detected by satellite gravity
missions like GRACE (the Gravity Recovery and Climate Experiment). The “GIA fingerprints”
constitute a standard output of SELEN^{4}. Along with the source code, we provide a supplementary
document with a full account of the theory, some numerical results obtained from a standard run,
and a user guide. Originally, the `SELEN`

program was conceived by Giorgio Spada (GS) in 2005 as a tool for students eager
to learn about GIA, and it has been the first SLE solver made available to the
community.

How to cite

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top
How to cite.

Spada, G. and Melini, D.: SELEN^{4} (SELEN version 4.0): a Fortran program for solving the gravitationally and topographically self-consistent sea-level equation in glacial isostatic adjustment modeling, Geosci. Model Dev., 12, 5055–5075, https://doi.org/10.5194/gmd-12-5055-2019, 2019.

1 Introduction

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In the last few decades, glacial isostatic adjustment (GIA) modeling has progressively gained a central role in the study of
contemporary sea-level change. Sea-level variations observed at the tide gauges
deployed along the world's coastlines need to be corrected for the effect of GIA
to assess the impact of global warming. As discussed in the review of
Spada and Galassi (2012), a precise estimate of global sea-level rise has been possible only
after Peltier and Tushingham (1989) first solved the sea-level equation (SLE) using an appropriate spatial resolution,
building upon the seminal papers of Farrell and Clark (1976) and Clark et al. (1978). Since then, a number of GIA models
characterized by different assumptions about the Earth's rheological profile and the history of
the Late Pleistocene ice sheets have been proposed, constrained by sea-level proxies
dating from the Last Glacial Maximum (LGM, 21 000 years ago).
For a
review of the development of GIA modeling, the reader is referred to Whitehouse (2009),
Spada (2017), and Whitehouse (2018).
GIA models have provided increasingly
accurate estimates of global mean secular sea-level rise (a summary is given in Table 1 of Spada and Galassi, 2012) but also have the potential of describing the patterns of future trends of sea level
in a global change scenario (see, e.g., Bamber et al., 2009; Spada et al., 2013).
Since the beginning of the “altimetry era” (1992–today) and the launch of the Gravity
Recovery and Climate Experiment (GRACE; see Wahr et al., 1998) in 2002, GIA modeling has
regained momentum, providing the tools for isolating the effects of global warming (i) from
absolute sea-level data (Nerem et al., 2010; Cazenave and Llovel, 2010) and (ii) from the
Stokes coefficients of the gravity field (see Leuliette and Miller, 2009; Cazenave et al., 2009; Chambers et al., 2010; WCRP, 2018)
to infer the ocean mass variation. Despite the GIA phenomenon now being tightly integrated into the science of global
change (Church et al., 2013), few efforts have been devoted so far to the development of open-source
codes for the solution of the SLE, although several post-glacial rebound simulators
(e.g., `TABOO`

; see Spada et al., 2004, 2011)
and Love number calculators have been made available to the
community (Spada, 2008; Melini et al., 2015; Bevis et al., 2016; Kachuck and Cathles, 2019).
As far as we know, the only publicly available and open-source
program in which the SLE is solved in its complete form is `SELEN`

(SealEveL EquatioN solver).
The SLE solver ISSM-SESAW v1.0 (Ice Sheet System Model – Solid Earth and Sea-level
Adjustment Workbench; Adhikari et al., 2016), being oriented to short-term cryosphere and
climate changes, only accounts for the elastic deformation of the Earth.
The open-source SLE solver `giapy`

(Kachuck, 2017; Martinec et al., 2018),
available from https://github.com/skachuck/giapy (last access: 26 November 2019), can deal with
complex ice models and viscoelastic rheology but does not take rotational
effects into account.

`SELEN`

was first presented to the GIA modeling community by Spada and Stocchi (2007),
who numerically implemented the SLE theory reviewed in Spada and Stocchi (2006).
`SELEN`

was fully based on the classical formulation of Farrell and Clark (1976); hence, the fixed-shoreline
approximation was assumed, and no account was given for rotational effects on sea-level variations. `SELEN`

used the Love number calculator `TABOO`

(see Spada et al., 2011) as a subroutine
and was tied to the Generic Mapping Tools (GMT; see Wessel and Smith, 1998) for the construction
of the present-day ocean function. In `SELEN`

and in all its subsequent versions, the
numerical integration of the SLE over the sphere takes advantage of the icosahedron-based pixelization
proposed by Tegmark (1996). Similarly, all the versions are based upon the pseudo-spectral method
of Mitrovica and Peltier (1991) and Mitrovica et al. (1994) for the solution of the SLE.
Originally, `SELEN`

came without a user guide, and it was disseminated via email by the authors. After `SELEN`

was first published in 2007, a number of improvements were made in terms of computational efficiency, portability,
and versatility but left the physical ingredients of the original code unaltered. This led to a new version of
the program, named SELEN 2.9 and announced by Spada et al. (2012). Since 2015,
the Computational Infrastructure for Geodynamics (CIG; http://www.geodynamics.org/, last access: 26 November 2019) has been hosting
SELEN 2.9 in its
short-term crustal dynamics section (see http://geodynamics.org/cig/software/selen, last access: 26 November 2019), from
where it can be freely downloaded along with a theory booklet and a fully detailed user
guide (Spada and Melini, 2015). Since
the year 2012, with the aid of Florence Colleoni and thanks to the feedback of a number of colleagues and
students, Giorgio Spada (GS)
and Daniele Melini (DM) have implemented new modules aimed at solving the SLE in the presence of rotational
effects and taking the migration of the shorelines into account. This has progressively led to several interim versions
of the program (SELEN 3.*x*), which have been tested intensively and validated during the years but never
officially released. We note that building upon `SELEN`

, some colleagues have independently developed
other versions of the code aimed at specific tasks, such as the study of the coupling between the SLE and ice
dynamics (de Boer et al., 2017).

Taking advantage of the experience developed since `SELEN`

was first
designed, we are now publishing a new version of the code named SELEN^{4} (SELEN version 4.0).
With respect to previous versions, SELEN^{4} has been improved in several
aspects. (i) The underlying SLE theory has been fully revised and now it
accounts both for horizontal migration of shorelines and for rotational
effects, resulting in a more realistic description of the GIA processes.
(ii) The package has been streamlined and reorganized into two
independent modules: a solver, which obtains a numerical
solution of the SLE in the spectral domain, and a
post-processor, which computes a full suite of observable
quantities through a spherical harmonic synthesis. This new structure facilitates
code portability, reusability, and customization, enabling the adaptation of SELEN^{4}
to new use cases. (iii) The SELEN^{4} modules have been completely
rewritten using symbol names that closely match those of the variables introduced in this
paper for the ease of code readability. Particular attention has been
paid to the optimization of the SLE solver, resulting in a large extent
of shared-memory parallelism, which allows for an efficient scaling to high
resolutions on multi-core systems.
(iv) SELEN^{4} has been
decoupled from the GMT software package, which
is no longer strictly required to run a GIA simulation, thus facilitating
code portability on high-performance systems where GMT may not be
available. SELEN^{4} still takes advantage of GMT (version 4) to produce various
graphical outputs through plotting scripts included in the distribution
package. (v) SELEN^{4} no longer calls the post-glacial rebound solver `TABOO`

as an internal
subroutine to compute the viscoelastic loading and tidal Love numbers, which are instead
supplied by the user through a data file. In this way, any set of Love
numbers can be used in SELEN^{4},
possibly overcoming some of the intrinsic limitations of
existing Love number calculators like `TABOO`

.
(vi) Recently, a prototype version of SELEN^{4} has been
successfully validated in a community benchmark of independently developed GIA codes
(Martinec et al., 2018). In the benchmark tests, simplified surface
loads have been employed, and the rotational effects have been ignored. In a further
benchmark effort, we are planning to adopt a realistic description
of the surface load and the contribution from the Earth's rotation.

The paper is organized into three main sections. In Sect. 2, we present a condensed
theory background for the SLE. In Sect. 3,
we describe the outputs obtained by a standard, intermediate-resolution run of SELEN^{4}. In Sect. 4,
we draw our conclusions.

2 Theory

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Here, we obtain the SLE from first principles, leaving a number of details of the theory to the
supplementary material, hereafter referred to as SSM19. The focus is on the various forms
the SLE takes, which are characterized by increasing complexity; the goal is to
obtain a formulation suitable for numerical discretization, which is given and
analyzed in SSM19 and implemented in SELEN^{4}. In an attempt to simplify the presentation,
and to obtain compact expressions for all the quantities involved, we are not exactly following the traditional
notation adopted in the literature since the seminal paper by Farrell and Clark (1976), hereafter referred to as FC76.
The number of definitions and variables involved in the construction of the SLE is remarkable and some
derivations are cumbersome; to facilitate the readers – especially those who are approaching the GIA
problem for the first time – we refer to the synopsis presented in SSM19 (see Table S8).
The SELEN^{4} program is written in a plain way, adopting constants and variables names that follow the same
notation employed in this theory section and in SSM19, with the aim of facilitating code readability.

We consider the system composed by the ice and the water in the oceans at a given
time *t*. Its mass can be expressed as

$$\begin{array}{}\text{(1)}& M\left(t\right)=\underset{e}{\int}L\mathrm{d}A,\end{array}$$

where *L*(*γ*,*t*) is the surface load; *γ* stands for (*θ*,*λ*), where *θ*
and *λ* are the geocentric colatitude and longitude, respectively; the integral is over the
whole Earth's surface; and d*A*=*a*^{2}sin *θ*d*θ*d*λ* is the area element, *a* being the
average radius of the Earth. According to Eq. (1), *L*
represents the mass per unit area distributed over the Earth's
surface:

$$\begin{array}{}\text{(2)}& L(\mathit{\gamma},t)={\displaystyle \frac{\mathrm{d}M}{\mathrm{d}A}},\end{array}$$

with $M\left(t\right)={M}^{\mathrm{i}}+{M}^{\mathrm{w}}$, where *M*^{i} and *M*^{w} are the mass
of the ice and the water at time *t*.
By obtaining explicit expressions for *M*^{i} and *M*^{w}, in SSM19, we
show that *L* can be expressed as the sum of two contributions, which account
(i) for the load exerted by the grounded ice and (ii) for the load of water on the oceans' floors, respectively:

$$\begin{array}{}\text{(3)}& L(\mathit{\gamma},t)={\mathit{\rho}}^{\mathrm{i}}IC+{\mathit{\rho}}^{\mathrm{w}}BO,\end{array}$$

where *ρ*^{i} and *ρ*^{w} are the ice and ocean water densities, *I* is the ice thickness,
*B* is sea level (i.e., the offset between the sea surface and the surface of the solid Earth),
$C=\mathrm{1}-O$ is the continent function (CF), where the ocean function (OF) is

$$\begin{array}{}\text{(4)}& {\displaystyle}O(\mathit{\gamma},t)\equiv \left\{\begin{array}{ll}\mathrm{1},& \text{if}\phantom{\rule{0.25em}{0ex}}T+\frac{{\mathit{\rho}}^{\mathrm{i}}}{{\mathit{\rho}}^{\mathrm{w}}}I<\mathrm{0}\\ \mathrm{0},& \text{if}\phantom{\rule{0.25em}{0ex}}T+\frac{{\mathit{\rho}}^{\mathrm{i}}}{{\mathit{\rho}}^{\mathrm{w}}}I\ge \mathrm{0}.\end{array}\right.\end{array}$$

Following Kendall et al. (2005) (see their Eq. 2), we have defined bedrock topography *T* as the negative
of the globally defined sea level:

$$\begin{array}{}\text{(5)}& T(\mathit{\gamma},t)=-B.\end{array}$$

We remark that
due to the horizontal migration of the shorelines and to the transition between floating and grounded ice,
*O* and *C* are, in general, time dependent.

In the following, we are concerned with time variations of the fields involved in the SLE,
which shall be denoted by calligraphic capital letters. These variations are
relative to a reference equilibrium state, established for $t\le {t}_{\mathrm{0}}\equiv \mathrm{0}$, during which all fields have a constant value.
Conventionally, time *t*=*t*_{0} denotes the time at which the
surface load changes (see, e.g., Kendall et al., 2005).
Accordingly, using Eq. (3)
in Eq. (1), the mass variation $\mathcal{M}\left(t\right)=M-{M}_{\mathrm{0}}$ of the system composed by ice and water
is

$$\begin{array}{}\text{(6)}& \mathcal{M}\left(t\right)={\mathit{\rho}}^{\mathrm{i}}\underset{e}{\int}\left(IC-{I}_{\mathrm{0}}{C}_{\mathrm{0}}\right)\mathrm{d}A+{\mathit{\rho}}^{\mathrm{w}}\underset{e}{\int}\left(BO-{B}_{\mathrm{0}}{O}_{\mathrm{0}}\right)\mathrm{d}A,\end{array}$$

where subscript 0 denotes reference quantities (i.e., constant values attained
for *t*≤0), and the first term on the right-hand side
represents the mass variation of the grounded ice, which we denote by *μ*(*t*).
Since mass must be conserved, we have

$$\begin{array}{}\text{(7)}& \mathcal{M}\left(t\right)=\underset{e}{\int}\mathcal{L}\mathrm{d}A=\mathrm{0},\end{array}$$

where

$$\begin{array}{}\text{(8)}& \mathcal{L}(\mathit{\gamma},t)\equiv L-{L}_{\mathrm{0}}.\end{array}$$

is the surface load variation. From Eq. (7), it follows immediately that an equivalent form of the mass conservation constraint is

$$\begin{array}{}\text{(9)}& <\mathcal{L}{>}^{\mathrm{e}}\left(t\right)=\mathrm{0}\phantom{\rule{0.125em}{0ex}},\end{array}$$

where $<\mathrm{\dots}{>}^{\mathrm{e}}$ indicates the average over the whole Earth's surface. Hereinafter, we shall consider only plausible surface loads, which we define here as those loads for which the mass is conserved. A discussion on how these surface loads can be realized is given by Bevis et al. (2016), who however have adopted ad hoc assumptions on how the mass lost from the ice sheets is redistributed. The SLE, however, does not rely upon such assumptions, since it defines a gravitationally self-consistent and non-uniform mass distribution over the oceans.

In SSM19, a suitable decomposition is found for ℒ, namely

$$\begin{array}{}\text{(10)}& \mathcal{L}(\mathit{\gamma},t)={\mathcal{L}}^{\mathrm{a}}+{\mathcal{L}}^{\mathrm{b}}+{\mathcal{L}}^{\mathrm{c}},\end{array}$$

where the first term,

$$\begin{array}{}\text{(11)}& {\mathcal{L}}^{\mathrm{a}}(\mathit{\gamma},t)={\mathit{\rho}}^{\mathrm{i}}C\mathcal{I},\end{array}$$

is associated with the ice thickness variation:

$$\begin{array}{}\text{(12)}& \mathcal{I}(\mathit{\gamma},t)=I-{I}_{\mathrm{0}};\end{array}$$

the second,

$$\begin{array}{}\text{(13)}& {\mathcal{L}}^{\mathrm{b}}(\mathit{\gamma},t)={\mathit{\rho}}^{\mathrm{w}}O\mathcal{S},\end{array}$$

is associated with sea-level change:

$$\begin{array}{}\text{(14)}& \mathcal{S}(\mathit{\gamma},t)=B-{B}_{\mathrm{0}};\end{array}$$

and the third,

$$\begin{array}{}\text{(15)}& {\mathcal{L}}^{\mathrm{c}}(\mathit{\gamma},t)={\mathit{\rho}}^{\mathrm{r}}Q\mathcal{O},\end{array}$$

is associated with OF variations:

$$\begin{array}{}\text{(16)}& \mathcal{O}(\mathit{\gamma},t)=O-{O}_{\mathrm{0}},\end{array}$$

where *ρ*^{r} is an arbitrary reference density and *Q* a time-invariant
auxiliary variable. We note that, by the same definition of OF, its variation
𝒪(*γ*,*t*) can only take
the values $(-\mathrm{1},\mathrm{0},+\mathrm{1})$.

The sea-level equation expresses the relationship between sea-level change, the sea surface variation, and the vertical displacement of the solid Earth. In this section, various forms of this relationship are described.

Above, sea level *B* has been qualitatively defined as the offset between the sea surface and the surface of the solid
Earth. More specifically,
denoting by *r*^{ss}(*γ*,*t*) and *r*^{se}(*γ*,*t*) the radii of the sea surface and of
the Earth's solid surface in a geocentric reference frame, respectively, sea level can
be equivalently expressed as

$$\begin{array}{}\text{(17)}& B(\mathit{\gamma},t)={r}^{\mathrm{ss}}-{r}^{\mathrm{se}},\end{array}$$

and in the reference state,

$$\begin{array}{}\text{(18)}& {B}_{\mathrm{0}}\left(\mathit{\gamma}\right)={r}_{\mathrm{0}}^{\mathrm{ss}}-{r}_{\mathrm{0}}^{\mathrm{se}}.\end{array}$$

So, introducing the sea surface variation,

$$\begin{array}{}\text{(19)}& \mathcal{N}(\mathit{\gamma},t)={r}^{\mathrm{ss}}-{r}_{\mathrm{0}}^{\mathrm{ss}},\end{array}$$

and the vertical displacement of the solid surface of the Earth,

$$\begin{array}{}\text{(20)}& \mathcal{U}(\mathit{\gamma},t)={r}^{\mathrm{se}}-{r}_{\mathrm{0}}^{\mathrm{se}},\end{array}$$

using Eq. (14), we obtain the SLE in its most basic form:

$$\begin{array}{}\text{(21)}& \mathcal{S}(\mathit{\gamma},t)=\mathcal{N}-\mathcal{U},\end{array}$$

expressing the variation of the height of the water column bounded by the sea surface and the ocean floor.

The sea surface variation is tightly associated with variations in the Earth's gravity field. Indeed, FC76 have shown that

$$\begin{array}{}\text{(22)}& \mathcal{N}(\mathit{\gamma},t)=\mathcal{G}+c,\end{array}$$

where the displacement of the geoid is obtained by the Bruns formula:

$$\begin{array}{}\text{(23)}& \mathcal{G}(\mathit{\gamma},t)={\displaystyle \frac{\mathrm{\Phi}-{\mathrm{\Phi}}_{\mathrm{0}}}{g}},\end{array}$$

(Heiskanen and Moritz, 1967) where Φ
is gravity potential, which includes
both the effects of surface loading and those from rotational variations (Martinec and Hagedoorn, 2014),
*g* is the reference gravity acceleration at the Earth's surface,
and *c* is a spatially invariant term introduced by FC76 to ensure mass conservation. The *c* term
is known within the GIA community as the FC76 constant, and its physical origin is explained in
detail by Tamisiea (2011).
Note that as a consequence of Eq. (22), the change in the sea surface elevation does not coincide
with the variation of the geoid elevation. Hence, using Eq. (22), the SLE (21) becomes

$$\begin{array}{}\text{(24)}& \mathcal{S}(\mathit{\gamma},t)=\mathcal{R}+c,\end{array}$$

where

$$\begin{array}{}\text{(25)}& \mathcal{R}(\mathit{\gamma},t)=\mathcal{G}-\mathcal{U}\end{array}$$

shall be referred to as sea-level response function.

We now assume that the responses to surface loading and to changes in the centrifugal potential can be combined linearly. Accordingly, the SLE (Eq. 24) can be further rearranged as

$$\begin{array}{}\text{(26)}& \mathcal{S}(\mathit{\gamma},t)={\mathcal{R}}^{\mathrm{sur}}+c+{\mathcal{R}}^{\mathrm{rot}},\end{array}$$

where

$$\begin{array}{}\text{(27)}& {\mathcal{R}}^{\mathrm{sur}}(\mathit{\gamma},t)={\mathcal{G}}^{\mathrm{sur}}-{\mathcal{U}}^{\mathrm{sur}}\end{array}$$

and

$$\begin{array}{}\text{(28)}& {\mathcal{R}}^{\mathrm{rot}}(\mathit{\gamma},t)={\mathcal{G}}^{\mathrm{rot}}-{\mathcal{U}}^{\mathrm{rot}}\end{array}$$

are the surface and the rotation sea-level response functions, whereas

$$\begin{array}{}\text{(29)}& \mathcal{G}(\mathit{\gamma},t)={\mathcal{G}}^{\mathrm{sur}}+{\mathcal{G}}^{\mathrm{rot}}\end{array}$$

and

$$\begin{array}{}\text{(30)}& \mathcal{U}(\mathit{\gamma},t)={\mathcal{U}}^{\mathrm{sur}}+{\mathcal{U}}^{\mathrm{rot}}\end{array}$$

are the geoid and the vertical displacement response functions, respectively.

By the constraint of mass conservation given by Eq. (7), the FC76 constant is easily determined. Using the expression found in SSM19 (see, in particular, Eq. S52 in the Supplement), the SLE (Eq. 26) becomes

$$\begin{array}{}\text{(31)}& \begin{array}{rl}\mathcal{S}(\mathit{\gamma},t)=& {\mathcal{S}}^{\mathrm{ave}}+\left({\mathcal{R}}^{\mathrm{sur}}-<{\mathcal{R}}^{\mathrm{sur}}{>}^{\mathrm{o}}\right)\\ & +\left({\mathcal{R}}^{\mathrm{rot}}\right.\left.-<{\mathcal{R}}^{\mathrm{rot}}{>}^{\mathrm{o}}\right),\end{array}\end{array}$$

where $<\mathrm{\dots}{>}^{\mathrm{o}}$
indicates the average over the (time-dependent) ocean surface defined by *O*=1, and

$$\begin{array}{}\text{(32)}& {\mathcal{S}}^{\mathrm{ave}}\left(t\right)={\mathcal{S}}^{\mathrm{equ}}+{\mathcal{S}}^{\mathrm{ofu}},\end{array}$$

where 𝒮^{equ} and 𝒮^{ofu} are two spatially invariant terms.
The first, referred to as equivalent sea-level change, is

$$\begin{array}{}\text{(33)}& {\mathcal{S}}^{\mathrm{equ}}\left(t\right)\equiv -{\displaystyle \frac{\mathit{\mu}}{{\mathit{\rho}}^{\mathrm{w}}{A}^{\mathrm{o}}}},\end{array}$$

where *μ* is the mass variation of the grounded
ice and *A*^{o} is the area of the oceans. The second,

$$\begin{array}{}\text{(34)}& {\mathcal{S}}^{\mathrm{ofu}}\left(t\right)\equiv {\displaystyle \frac{\mathrm{1}}{{A}^{\mathrm{o}}}}\underset{e}{\int}{T}_{\mathrm{0}}\mathcal{O}\mathrm{d}A,\end{array}$$

depends explicitly upon variations of the OF, either due to the horizontal migration of the shorelines or to
transitions from grounded to floating ice (or vice versa). It is important to note that
𝒮^{equ}(*t*) and 𝒮^{ofu}(*t*) are both model dependent via *A*^{o} and 𝒪.
Evaluating the ocean average of
both sides of Eq. (31), and observing
that $<<\mathcal{R}{>}^{\mathrm{o}}{>}^{\mathrm{o}}=<\mathcal{R}{>}^{\mathrm{o}}$ and
$<{\mathcal{S}}^{\mathrm{ave}}{>}^{\mathrm{o}}={\mathcal{S}}^{\mathrm{ave}}$, it is easily verified that
𝒮^{ave} simply represents the ocean-averaged relative
sea-level change:

$$\begin{array}{}\text{(35)}& {\mathcal{S}}^{\mathrm{ave}}\left(t\right)\equiv <\mathcal{S}{>}^{\mathrm{o}}.\end{array}$$

As a consequence of that, from Eq. (31), we see that the regional imprint of GIA on relative
sea-level change is totally determined by the response functions ℛ^{sur} and ℛ^{rot}.
We note that according to the analysis of Mitrovica and Milne (2002), the ocean-averaged terms
in Eq. (31) largely incorporate the far-field “ocean syphoning” effect,
which describes the flux of water away from the equatorial regions to fill the space left by the collapse of forebulges at the
periphery of previously glaciated regions.

In the classical FC76 framework, a constant OF is assumed and the effects arising from the Earth's
rotation are neglected. Hence, ℛ^{rot}=0 and 𝒪=0, with the latter condition
implying 𝒮^{ofu}=0 because of Eq. (34). In this context, the
SLE simply reduces to

$$\begin{array}{}\text{(36)}& {\mathcal{S}}^{\text{FC76}}(\mathit{\gamma},t)={\mathcal{S}}^{\mathrm{eus}}+\left({\mathcal{R}}^{\mathrm{sur}}-<{\mathcal{R}}^{\mathrm{sur}}{>}^{\mathrm{o}}\right),\end{array}$$

where

$$\begin{array}{}\text{(37)}& {\mathcal{S}}^{\mathrm{eus}}\left(t\right)=-{\displaystyle \frac{\mathit{\mu}}{{\mathit{\rho}}^{\mathrm{w}}{A}^{\mathrm{op}}}}\end{array}$$

is often referred to as “eustatic sea-level change”, and *A*^{op} represents the present-day area
of the surface of the oceans. Hence, for a rigid and non-gravitating Earth (for which ℛ^{sur}=0),
𝒮^{eus} would represent the (spatially invariant) relative sea-level change.
Note that 𝒮^{eus}, which only depends on the history of the past grounded ice
volume (Milne and Mitrovica, 2008),
should not be confused with 𝒮^{ave}, given by Eq. (32), since the latter is dynamically dependent
upon the Earth's response through *A*^{o}(*t*) and 𝒪(*t*) (e.g., Spada, 2017).
We note that in the current terminology for sea level (Gregory et al., 2019),
the term “barystatic” should be preferred to “eustatic”, a term
originally attributed to Suess (1906) and that has since then received many qualifications
(see FC76, p. 648). Hence, it would be more correct to write

$$\begin{array}{}\text{(38)}& {\mathcal{S}}^{\mathrm{bar}}\left(t\right)=-{\displaystyle \frac{\mathit{\mu}}{{\mathit{\rho}}^{\mathrm{w}}{A}^{\mathrm{op}}}}.\end{array}$$

Following, e.g., Milne and Mitrovica (1998), the surface response function ℛ^{sur}
in Eq. (31) is obtained by a 3-D spatiotemporal convolution:

$$\begin{array}{}\text{(39)}& {\mathcal{R}}^{\mathrm{sur}}(\mathit{\gamma},t)={\mathrm{\Gamma}}^{\mathrm{s}}\otimes \mathcal{L},\end{array}$$

where Γ^{s}(*γ*,*t*) is the surface sea-level Green function. The details of the expansion
of ℛ^{sur} in series of spherical harmonics are somewhat cumbersome and can be found in
SSM19. Here, we only note that using Eq. (10) we have ${\mathcal{R}}^{\mathrm{sur}}={\mathcal{R}}^{\mathrm{a}}+{\mathcal{R}}^{\mathrm{b}}+{\mathcal{R}}^{\mathrm{c}}$, where the three terms are obtained by convolving Γ^{s}
with ℒ^{a}, ℒ^{b}, and ℒ^{c},
respectively. Contrary to ℛ^{sur}, the harmonic coefficients of the
rotational response ℛ^{rot}(*γ*,*t*) are directly obtained by a 1-D time convolution:

$$\begin{array}{}\text{(40)}& {\mathcal{R}}_{lm}^{\mathrm{rot}}\left(t\right)={\mathrm{{\rm Y}}}_{l}^{\mathrm{s}}\cdot {\mathrm{\Lambda}}_{lm},\end{array}$$

where ${\mathrm{{\rm Y}}}_{l}^{\mathrm{s}}\left(t\right)$ is the rotation sea-level Green function and Λ_{lm}(*t*) is the
coefficients of degree *l* and order *m* of the expansion of the variation of centrifugal potential
Λ(*γ*,*t*)
associated with changes in the
Earth's angular velocity (Milne and Mitrovica, 1998). In SSM19, it is shown that
Λ(*γ*,*t*) is essentially a spherical harmonic function
of degree *l*=2 and order $m=\pm \mathrm{1}$, with the other degree *l*=2 terms and the degree *l*=0 in the expansion of
Λ(*γ*,*t*) being negligible (see also Martinec and Hagedoorn, 2014).

Thus, the SLE (31) can be rearranged as

$$\begin{array}{}\text{(41)}& \mathcal{S}(\mathit{\gamma},t)={\mathcal{S}}^{\mathrm{ave}}+{\mathcal{R}}^{\prime \phantom{\rule{0.25em}{0ex}}\mathrm{a}}+{\mathcal{R}}^{\prime \phantom{\rule{0.25em}{0ex}}\mathrm{b}}+{\mathcal{R}}^{\prime \phantom{\rule{0.25em}{0ex}}\mathrm{c}}+{\mathcal{R}}^{\prime \phantom{\rule{0.25em}{0ex}}\mathrm{rot}},\end{array}$$

where the primed response functions are

$$\begin{array}{}\text{(42)}& {\displaystyle}& {\displaystyle}{\mathcal{R}}^{\prime \phantom{\rule{0.25em}{0ex}}\mathrm{abc}}(\mathit{\gamma},t)={\mathcal{R}}^{\mathrm{abc}}-<{\mathcal{R}}^{\mathrm{abc}}{>}^{\mathrm{o}},\text{(43)}& {\displaystyle}& {\displaystyle}{\mathcal{R}}^{\prime \phantom{\rule{0.25em}{0ex}}\mathrm{rot}}(\mathit{\gamma},t)={\mathcal{R}}^{\mathrm{rot}}-<{\mathcal{R}}^{\mathrm{rot}}{>}^{\mathrm{o}}.\end{array}$$

We note that ℛ^{′ b} depends on *O*𝒮 through the surface
load variation ℒ^{b} (see Eq. 13); in SSM19, it is shown
that this holds for ℛ^{rot} as well. Following Mitrovica and Peltier (1991),
in view of the numerical solution of the SLE, it is therefore convenient
to transform Eq. (41) in such a way that *O*𝒮 becomes the
unknown in lieu of 𝒮. This is accomplished by projecting Eq. (41) on
the OF (i.e., multiplying both sides of the SLE by *O*), which provides the final form
of the SLE:

$$\begin{array}{}\text{(44)}& \mathcal{Z}(\mathit{\gamma},t)={\mathcal{Z}}^{\mathrm{ave}}+{\mathcal{K}}^{\mathrm{a}}+{\mathcal{K}}^{\mathrm{b}}\left(\mathcal{Z}\right)+{\mathcal{K}}^{\mathrm{c}}+{\mathcal{K}}^{\mathrm{rot}}\left(\mathcal{Z}\right),\end{array}$$

where

$$\begin{array}{}\text{(45)}& \mathcal{Z}(\mathit{\gamma},t)=O\mathcal{S}\phantom{\rule{0.125em}{0ex}},\end{array}$$

and

$$\begin{array}{}\text{(46)}& {\displaystyle}& {\displaystyle}{\mathcal{Z}}^{\mathrm{ave}}(\mathit{\gamma},t)=O{\mathcal{S}}^{\mathrm{ave}}=O\left({\mathcal{S}}^{\mathrm{equ}}+{\mathcal{S}}^{\mathrm{ofu}}\right)\phantom{\rule{0.125em}{0ex}},\text{(47)}& {\displaystyle}& {\displaystyle}{\mathcal{K}}^{\mathrm{abc}}(\mathit{\gamma},t)=O{\mathcal{R}}^{\prime \phantom{\rule{0.25em}{0ex}}\mathrm{abc}}\phantom{\rule{0.125em}{0ex}},\text{(48)}& {\displaystyle}& {\displaystyle}{\mathcal{K}}^{\mathrm{rot}}(\mathit{\gamma},t)=O{\mathcal{R}}^{\prime \phantom{\rule{0.25em}{0ex}}\mathrm{rot}}.\end{array}$$

The dependence of 𝒦^{b} and 𝒦^{rot}
upon 𝒵 in Eq. (44) manifests the implicit nature of the SLE, which is a
3-D non-linear integral equation, similar, in some respect, to an inhomogeneous
and non-linear Fredholm equation of the second
kind (e.g., Jerri, 1999; Spada, 2017). For the spectral
discretization of Eq. (44) and for a detailed illustration of the
scheme adopted to solve the SLE, the reader is referred to Sects. S7 and S8.7 of SSM19. Here, it is only worth mentioning that
the SLE is solved by a pseudo-spectral iterative approach (Mitrovica and Peltier, 1991; Mitrovica and Milne, 2003) complemented
by a spatial discretization on an icosahedron-based spherical geodesic grid (Tegmark, 1996). Two nested
iterations are adopted, where the external one refines the topography progressively
using the solution of the SLE obtained in the internal one.

3 A test run with SELEN^{4}

Back to toptop
In the following, we illustrate some of the outputs of a standard SELEN^{4} run in which the resolution of the
Tegmark grid is set to *R*=44 (see Sect. S8.6 in the Supplement), while the maximum harmonic degree of the spectral
decomposition is ${l}_{max}=\mathrm{128}$.
Note that condition $\mathit{\text{P}}\ge \phantom{\rule{0.25em}{0ex}}{l}_{max}^{\mathrm{2}}/\mathrm{3}$, where $P=\mathrm{40}R(R-\mathrm{1})+\mathrm{12}$ is the number of pixels in the grid
for a given *R* value, must be necessarily met to preserve the properties of the spherical harmonics on the grid
as discussed in Sect. S8.6 and in Tegmark (1996).
In the test run, we solve the SLE by three external and three internal iterations (${n}_{\mathrm{ext}}={n}_{\mathrm{int}}=\mathrm{3}$;
see Sect. S8.7),
as customarily adopted in GIA studies as in Kendall et al. (2005).
Henceforth, the notation *R44/L128/I3* shall be employed to denote these fundamental SELEN^{4} settings. Of
course, with increasing values of parameters (*R*,
*l*_{max}, *n*_{ext}, *n*_{int}), more accurate results are expected, which however might come
with a substantial increase in the computational burden. In the SELEN^{4} test run, we
account for both loading and rotational effects.

We assume that the user has installed and executed the program following the indications given in
the user guide of SELEN^{4}. Most of the program outputs discussed in this section have been obtained using
the same configuration file that comes with the SELEN^{4} package. However, some results shall be based on
different settings, in order to appreciate the sensitivity of the outputs on key configuration parameters. We
first describe the GIA model adopted in the test run, which consists of three elements, i.e., an ice
melting history, a description of the present-day global relief, and a 1-D rheological model of the Earth's
mantle. Then, browsing the output folders of SELEN^{4}, we illustrate and discuss two distinct output sets, pertaining
to the past and present effects of GIA on sea-level change and on geodetic variations, respectively.
The features of the test run are summarized in Table 1. For
reference, on a 12-core mid-2012 Mac Pro, the execution time of this test run of SELEN^{4} is 1 h and 15 min.

In principle, there are no restrictions on the spatial and temporal features of the ice melting history that can
be employed in SELEN^{4}, provided that the model is properly discretized according to the scheme outlined in
Sect. S8.7. Similarly, any linear rheological profile is a priori acceptable for
the mantle and for the lithosphere, as long as the Love numbers can be cast in a normal-mode multi-exponential
form (Peltier, 1974).
Due to its central role in the context of contemporary GIA studies, in our test run, we have implemented an
ad hoc realization of the GIA model known as ICE-6G_C (VM5a), originally introduced by Peltier et al. (2015).

Ice thickness data for model ICE-6G_C have been downloaded from the home
page of William R. Peltier in August 2016. The data span
the last 26 000 years and are provided on a $\mathrm{1}{}^{\circ}\times \mathrm{1}{}^{\circ}$
global latitude–longitude grid (the number of grid points is thus 64 800). In each of the grid cells, the time history
of ice thickness is assumed to evolve in a piecewise linear manner, with variable increments of 1.0 or 0.5 kyr. Thus,
to fit the SELEN^{4} default input format, we have first mapped the original thickness data on a spherical
equal-area Tegmark grid described in Sect. S8.6 of SSM19. In doing that, we have chosen a resolution
parameter *R*=44, so that the grid consists of *P*=75 692 pixels (or cells), each with a radius of ∼46 km.
The cells' number is thus a bit larger than the number of cells in the original grid.
In addition, we have transformed the original time history in a piecewise constant form with a uniform
spacing of 0.5 kyr, assuming no glaciation phase prior to deglaciation. Because of the adaptations we have made,
the ice model obtained is not an exact replica of ICE-6G_C but a particular realization of it. Hence, to
avoid any ambiguity, in the following, it shall be referred to as I6G-T05-R44. Assuming an ice density
*ρ*^{w}=931 kg m^{−3} and that the area of the oceans is fixed to the present value,
I6G-T05-R44 holds 206.5 m of
equivalent sea level at 26 ka and 75.1 m at present, corresponding to a total eustatic sea-level rise
of 131.4 m since the inception of melting (26 ka). The ice thickness of I6G-T05-R44 for a few time frames is shown
in Fig. 1.

In order to reconstruct the whole history of the Earth's topography and of relative sea level
(RSL)
since the inception of deglaciation, it is necessary to impose the present relief as a final condition
for the sea-level equation (see Peltier, 1994). In this test run, we have utilized
the “bedrock version” of the global ETOPO1 dataset (Amante and Eakins, 2009; Eakins and Sharman, 2012)
as the final condition, whereas the final condition for the ice thickness is given by the last time frame
of I6G-T05-R44. ETOPO1
is distributed on a longitude–latitude grid with a resolution of 1 arcmin ($\mathrm{1}/\mathrm{60}{}^{\circ}$), so that
an interpolation of topography on the Tegmark grid has been performed.
Note that in regions where small-scale topographic features are present, such as ocean trenches, interpolating
a high-resolution relief model on a coarse Tegmark grid may yield inaccurate results. In this case,
a different approach should be adopted for the realization of topography, for instance, based on a binning algorithm.
Other choices of the final
topography are of course possible. For example, in order to ensure the maximum accuracy in Antarctica,
the original model,
ICE-6G_C (VM5a), employs the Bedmap2 dataset of Fretwell et al. (2013) south of 60^{∘} S latitude.
In SELEN^{4}, there are no restrictions on the choice of the final relief, which is left to the user. In Fig. 2,
we show the realization of the modern bathymetry that we have obtained by interpolating ETOPO1 on the same
Tegmark grid that we have used for the ice sheets in this test run, which shall be referred to as model
ETO-R44 in the following. With SELEN^{4}, other versions of this elevation model are made available,
characterized by different spatial resolutions.

The 11-layer Maxwell rheological profile of the 1-D Earth employed in the test run is shown in
Table 2. For each layer, values of density and of rigidity are obtained by volume averaging the PREM
(Preliminary Reference Earth Model) of
Dziewonski and Anderson (1981), while the viscosity profile is reproduced using the data available in the
Supplement in Peltier et al. (2015). The 90 km thick lithosphere is elastic and the core
is fluid, homogeneous, and
inviscid. Note that the original VM5a profile includes elastic compressibility and reproduces the finely layered PREM
structure (Peltier et al., 2015); since we are employing an incompressible
rheology, in the following, we shall refer to the model in Table 2 as VM5i.
As it includes *N*_{v}=9 distinct Maxwell layers in the mantle, for any
given harmonic degree *l*, the loading Love numbers
(LLNs) and tidal Love numbers (TLNs) for the VM5i model are described
by a spectrum of 4*N*_{v}=36 viscoelastic normal modes (see, e.g., Spada et al., 2011).
Since we are not aware of published, community-agreed sets of Love
numbers for a multi-layered compressible viscoelastic model, in the test run we rest on an
incompressible profile, for which agreed results have been obtained by Spada et al. (2011). We remark,
however, that SELEN^{4} can work with compressible or transient rheologies as well,
provided that LLNs and TLNs in normal-mode form (Wu and Peltier, 1982) are accessible to the user.
Figure 3a and b show the elastic
and fluid values of the LLNs in the range of harmonic degrees $\mathrm{1}\le l\le \mathrm{1024}$ for the VM5i model.
The Love numbers are given in a geocentric reference frame
with the origin in the Earth’s
center of mass (CM).
Numerical values of a few relevant LLNs and TLNs are listed in Table 3
and in its caption.
They have been computed by the Love number calculator `TABOO`

(see Spada et al., 2011)
in a multi-precision environment (Smith, 1991; Spada, 2008).

This section is devoted to the description of some outputs of the SELEN^{4} test run, concerning
the effects of GIA during the whole period after the LGM. These include (i) the predictions of
the history of RSL at specific sites, (ii) the time evolution of
paleotopography in some regions of interest, and (iii) the excursions
of the Earth's pole of rotation forced by GIA.

Figure 4 shows data (with error bars) and SELEN^{4} predictions
for a small subset of the 392 sites contained in the RSL database of Tushingham and Peltier (1993),
hereafter referred to as TP93.
In view of its historical importance in the development of GIA
studies (e.g., Tushingham and Peltier, 1991, 1992; Melini and Spada, 2019),
the TP93 database is available with the SELEN^{4} package; however, there are no restrictions on the use
of other datasets, or simply individual RSL records, if available to the user.
Note that ages given in the TP93 database included with SELEN^{4} are expressed in radiocarbon
years and have not been calibrated to calendar years.
In Fig. 4, three different configurations, i.e., combinations of parameters
*R*, *l*_{max}, *n*_{ext}, and *n*_{int} have been adopted, always assuming
*n*_{ext}=*n*_{int}. Results obtained for the three
configurations are shown by different colors.
The black curves have been obtained using the GIA model described in Sect. 3.1,
characterized by the settings *R44/L128/I3* (*R*=44, *l*_{max}=128, ${n}_{\mathrm{ext}}={n}_{\mathrm{int}}=\mathrm{3}$).
Blue curves have been obtained by configuring SELEN^{4} with the combination of parameters
*R100/L512/I5*, i.e., increasing the spatial resolution and the number of internal and external iterations
in a significant way (a truncation degree ${l}_{max}=\mathrm{256}$ is often employed in GIA modeling; see,
e.g., Kendall et al., 2005). With this configuration, the pixel radius is reduced to ∼20 km
(see Sect. S8.6 and Table S6). Of course, the execution time of SELEN^{4} increases significantly with respect to the test
run, requiring 2.5 d (∼60 h) on a 56-core Intel Xeon E7 Broadwell system. It is apparent that this
high-resolution case is providing results substantially matching those of the standard run with *R44/L128/I3*.
Minor differences can be noted in the early stages of deglaciation, which however do not exceed the typical
uncertainty on the observed RSL values. These differences are likely to be caused by the significant changes that
the topography undergoes in this early phase in the polar regions, which are better captured by increasing the model
resolution. Finally, red curves have been obtained for a low-resolution run with *R30/L64/I2*, whose
execution time is 15 min on a 12-core mid-2012 Mac Pro. The curves clearly indicate that computationally inexpensive
runs can provide reliable results in the far field of the previously glaciated areas (e.g., at sites 639 and 525),
but in the near field (e.g., site 283) they can diverge significantly
from both high- and intermediate-resolution results.

From a visual inspection of Fig. 4, it is apparent that at some sites the
best GIA predictions fit very well the observations, like
at sites 101 and 283. For others, the trend of the RSL data is captured
satisfactorily (see sites 155, 209, and 328), while in some others the fit is quite poor
(sites 639, 525, and 570). The identification of the possible sources of the evidenced misfits,
which should be measured using rigorous statistical methods
after a proper calibration of the ages, is not
the purpose of this work. We only note that they do not necessarily stem from limitations of the GIA
model adopted, since it is well known that
at a specific site tectonic deformations can have an important
role (see, e.g., Antonioli et al., 2009, for a significant example),
and these are not taken into account when solving the SLE. Similarly, in our
formulation of the SLE, we are neglecting the possible effects from the loads exerted by
sediments (Dalca et al., 2013). Since the SELEN^{4} program is open source,
the users can modify the code to account for non-glacial loads
and change the configuration to determine more suitable combinations of the basic ingredients
of GIA modeling, i.e., the history of deglaciation and the rheological
layering of the mantle, in order to improve the fit between model predictions and any preferred RSL dataset.

SELEN^{4} allows for a gravitationally and topographically
self-consistent description of the evolution of sea level along the route highlighted by Peltier (1994)
and Lambeck (2004). This implies that SELEN^{4} can iteratively reconstruct the time evolution
of the coastlines and of the OF, in a fashion that is consistent with the gravitational, rotational, and
deformational effects induced
by deglaciation (for a full account of the theory, the reader is referred to SSM19). These
features make the SLE an integral 3-D non-linear equation (Spada, 2017).
The importance of the evolution of paleotopography for the development of human culture
since the LGM has been pointed out in a number of
works (see, e.g., Cavalli-Sforza et al., 1993; Peltier, 1994; Lambeck, 2004; Dobson, 2014, and references therein).
Recently, in the context of GIA modeling, Spada and Galassi (2017) have faced the problem of the dynamic
evolution of aquaterra, i.e., the land that has been inundated and exposed during the
last glacial cycle (Dobson, 1999, 2014), using the same approach adopted in this work.

A standard SELEN^{4} output is shown in Fig. 5,
where the Earth's relief at the LGM is reconstructed in
the post-processing phase of SELEN^{4}, according to the solution of the SLE. Note that the figure only shows the
bedrock relief at the LGM, which is not what Peltier (1994) has called true paleotopography
(PT), which also includes the contribution of ice elevation. The user can easily obtain maps of the
full PT merging maps like that shown in Fig. 5 with those of Fig. 1.
Light blue areas across the polar regions covered by thick ice at the LGM correspond to places where the ice was grounded
below sea level in that epoch. In SELEN^{4}, it is also possible to visualize the full time evolution of the OF
in order to better appreciate, for example, the spatiotemporal distribution of the ice shelves, which cannot
be seen in this paleotopography map.
At the global scale, the major land masses that were exposed at the LGM are clearly visible, as evidenced
by the low-elevation green areas; these include Beringia, Sundaland, Patagonia, Sahul, Doggerland,
and the East Siberian Arctic Shelf.
Using GIA models, the time evolution of these land masses has been investigated in an a number of
papers, also in view of the important implications on the dispersal of modern humans.
For specific case studies, the reader is referred to the works of Peltier and Drummond (2002),
Lambeck (2004), Klemann et al. (2015),
Spada and Galassi (2017), and references therein.

By increasing the spatial resolution, SELEN^{4} can also be employed to resolve the past sea-level variations
on a regional scale. As a specific case study, we consider the Mediterranean Sea.
The history of RSL in the Mediterranean Sea has been the subject of various investigations,
stimulated by the amount of high-quality geological, geomorphological, and archeological
indicators in the
region (see, e.g., Lambeck and Purcell, 2005; Antonioli et al., 2009; Evelpidou et al., 2012; Vacchi et al., 2016; Roy and Peltier, 2018, and references therein).
Since on the global scale of Fig. 5 the details of the paleotopography in this area are difficult to
visualize, we have used the outputs of the high-resolution run with settings *R100/L512/I5*, already
exploited in Sect. 3.2.1. The results are shown in the map of Fig. 6,
where paleotopography
is shown at 26 ka. The vastly exposed continental shelf of Tunisia (Mauz et al., 2015) and
the northern Adriatic Sea (Lambeck and Purcell, 2005) are now clearly visible, along with other
smaller-scale regions where the topography has seen significant changes during the last
deglaciation (Lambeck, 2004; Purcell and Lambeck, 2007).

With SELEN^{4}, three configurations are possible in which rotational effects on GIA are dealt
with in different manners. First, these effects can be simply ignored, as it is done in the classical
FC76 GIA theory. However, when rotational effects are taken into consideration, this can be done in two
different ways, i.e., either following the traditional rotation theory
(Milne and Mitrovica, 1998; Spada et al., 2011) or a revised rotation theory proposed
by Mitrovica et al. (2005) and Mitrovica and Wahr (2011).
The reader is referred to the literature for a detailed presentation of the two theories and
to Sect. S5.2 for a brief account. We also refer to Martinec and Hagedoorn (2014) for a general
formulation dealing with rotational effects in GIA modeling.
Here, it is useful to mention that in the traditional treatment the long-term response
of the Earth is evaluated assuming that the lithosphere is characterized by a finite elastic strength,
while in the revised theory the equilibrium rotational shape is, more realistically, only based on the viscous
properties of the planet. Furthermore, the long-term extra flattening due to mantle dynamics is properly
accounted for. We remark that in both cases the fast Chandler wobble component of polar motion is filtered out from the Liouville equations, because the timescales of GIA largely exceed the Chandler wobble
period (∼14 months; see, e.g., Lambeck, 1980).
The implications for the GIA response of the
new theory are quite significant, as illustrated in
detail by Mitrovica et al. (2005) and Mitrovica and Wahr (2011) and shown in
Fig. 7.

Solid curves in Fig. 7 show the evolution
of the polar motion components (*m*_{x},*m*_{y}) and their rates of change $({\dot{m}}_{x},{\dot{m}}_{y})$ in response to
GIA, obtained by solving the Liouville equations in the test run, in which the revised rotation theory is employed.
Dashed curves show results obtained with the traditional theory.
As the *x*
and *y* components of polar motion are conventionally measured
along the Greenwich meridian and 90^{∘} E, respectively, Fig. 7a indicates that
since the inception of deglaciation, the displacement of the pole has been roughly in the direction of the Hudson Bay,
consistent with the seminal results of Sabadini and Peltier (1981). The two theories predict similar evolutions
of the pole of rotation, which
according to Fig. 7a has been displaced by ∼18 km on the Earth's surface by
the glacial readjustment process since 26 ka. We note that at the time of the rapid melting episode known as
meltwater pulse 1A (MWP-1A, between 14.3 and 12.8 ka; see, e.g., Blanchon, 2011),
a sudden variation in the *m*_{y} component of polar motion had occurred but no changes could be observed
on *m*_{x}. When the rates of polar motion are considered in Fig. 7b,
differences between the predictions of the two rotations theories are more apparent. In particular, the present-day
(0 ka) rates of polar motion are found to be ∼1 and ∼3^{∘} Ma^{−1} for the revised and the
traditional theories, respectively, which fits the predictions of Mitrovica and Wahr (2011) and confirms that
the traditional theory largely overestimates the effects of GIA on polar motion. We note also that
MWP-1A has caused a remarkable acceleration of polar motion, with a rate as large of
∼5^{∘} Ma^{−1} near the end of the pulse; this value exceeds the rate of polar motion observed
since the year 1900
by a factor of ∼5 (see, e.g., Dickman, 1977).

In this section, we describe further outputs of the SELEN^{4} test run, focusing on the
effects of GIA at the present time. In particular, we consider (i) the global pattern
of the so-called GIA fingerprints, (ii) predictions of
the rate of sea-level change at tide gauges, and (iii) the time variations of the
Stokes coefficients of the Earth's gravity field induced by GIA.

Figure 8 shows another standard output of SELEN^{4},
i.e., the present-day rates of variation of four fundamental quantities associated with GIA,
obtained for the test run. These are relative sea-level change ($\dot{\mathcal{S}}$; Fig. 8a),
vertical displacement of the crust ($\dot{\mathcal{U}}$; Fig. 8b), absolute sea level
($\dot{\mathcal{N}}$; Fig. 8c), and
geoid height ($\dot{\mathcal{G}}$; Fig. 8d). Generalizing the definition of
sea-level fingerprint introduced by
Plag and Jüettner (2001) in the context of contemporary ice mass changes, these have been referred
to as GIA fingerprints by Spada and Melini (2019). The
spatial variability of the GIA fingerprints reflect the effects of deformation, gravitational attraction,
and rotation within the
system composed by the solid Earth, the oceans, and the ice sheets (Clark et al., 1978; Mitrovica and Milne, 2002).
In view of their importance on the interpretation of ground-based (King et al., 2010) or satellite
geodetic observations (Peltier, 2004) and of tide gauge secular
trends (e.g., Spada and Galassi, 2012; Wöppelmann and Marcos, 2016), their properties have been the
subject of various investigations
during last decade (see, e.g., Mitrovica et al., 2011; Tamisiea, 2011; Spada and Galassi, 2015; Spada, 2017; Husson et al., 2018; Melini and Spada, 2019; Spada and Melini, 2019).

It should be remarked that the four fingerprints shown in Fig. 8 are not independent of one another.
In particular, the SLE gives $\dot{\mathcal{S}}=\dot{\mathcal{N}}-\dot{\mathcal{U}}$ according to Eq. (21).
Furthermore, $\dot{\mathcal{N}}=\dot{\mathcal{G}}+\dot{c}$,
where *c* is the FC76 constant (see Sect. S2.4 in SSM19). The two relationships above
hold, regardless of the particular combination of
rheology and ice model employed, and the preferred rotation theory adopted. However, the patterns
of the GIA fingerprints and the numerical value of $\dot{c}$ are model dependent. Other interesting results hold for the
spatial averages of the fingerprints, which reflect some physical aspects of GIA (Spada, 2017; Spada and Melini, 2019) and
are useful to correct geodetic observations from the effects of deglaciation (e.g., Spada and Galassi, 2015).
In Table 4, we summarize the numerical
values of whole Earth surface averages (denoted by the symbol $<\mathrm{\cdots}{>}^{\mathrm{e}}$) and
ocean averages ($<\mathrm{\cdots}{>}^{\mathrm{o}}$) of the GIA fingerprints in the test run. In addition, we have
also executed SELEN^{4} adopting the traditional rotation theory and neglecting rotational effects,
and the corresponding averages are shown in Table 4 as well. We note that by virtue of mass
conservation, $<\dot{\mathcal{G}}{>}^{\mathrm{e}}=<\dot{\mathcal{U}}{>}^{\mathrm{e}}=\mathrm{0}$ (see Sects. S4.3 and S6.2), regardless
of the rotation theory adopted, and as a consequence $<\dot{\mathcal{S}}{>}^{\mathrm{e}}=<\dot{\mathcal{N}}{>}^{\mathrm{e}}=\dot{c}$.
We also note that
the numerical value of $<\dot{\mathcal{N}}{>}^{\mathrm{o}}$, commonly employed to correct
the altimetry observations of absolute sea-level change
for the effects of GIA, is in fair agreement with predictions from state-of-the-art GIA models
(e.g., Church et al., 2013; Spada and Galassi, 2015; Spada, 2017). Notably, $<\dot{\mathcal{N}}{>}^{\mathrm{o}}$ is
not very sensitive to the choice of the rotation theory. Since model I6G-T05-R44 assumes that melting
of the major ice sheets ceased ∼4000 years ago, the small value of $<\dot{\mathcal{S}}{>}^{\mathrm{o}}$ only
reflects ongoing changes in the area of the oceans due to GIA. In the FC76 fixed-shoreline approximation,
$<\dot{\mathcal{S}}{>}^{\mathrm{o}}$ would be identically zero by virtue of the mass conservation
principle (see, e.g., Spada, 2017; Spada and Melini, 2019).

Estimating global mean sea-level rise in response to climate change requires correcting tide gauge records for the effects of GIA. Since the late 1980s, with the awareness of global warming and the availability of numerical solutions of the SLE (Peltier and Tushingham, 1989), GIA corrections have been routinely applied to the observed trends of sea level (for a review, see Spada and Galassi, 2012; Spada et al., 2015; Wöppelmann and Marcos, 2016). However, since GIA models are progressively improved to provide a better description of reality, corrections at tide gauges are not given once and for all (Kendall et al., 2006; Tamisiea, 2011; Melini and Spada, 2019). Furthermore, new constraints from past sea level or modern geodetic observations have permitted gradual refinements (either by formal inverse methods or simply by trial and error) of the two basic ingredients of GIA modeling, i.e., the Earth rheological profile and the history of deglaciation since the LGM. Uncertainties in modeling are significant (Melini and Spada, 2019), which constitutes an additional motivation to improve the approach to GIA.

In Table 5, we show SELEN^{4} predictions for $\dot{\mathcal{S}}$ at a few tide gauges
for the test run and other possible configurations as well. Here, we only show results for the 23
sites that have been considered by Douglas (1997) in his redetermination of global
sea-level rise, which obey specific criteria that make them suitable to represent
the trend of secular sea-level rise. The sites chosen by Douglas (1997) are located in the periphery of the regions
covered by thick ice sheets at the LGM, since GIA predictions at sites formerly beneath the ice sheets
are expected to be more affected by uncertainties in GIA modeling. This has been recently
confirmed by Melini and Spada (2019). The post-processing phase of SELEN^{4} can
be configured to handle any properly formatted input dataset with coordinates
of geodetic points of interest, where all the variables considered in Fig. 8 can be evaluated.
This can be useful, for instance, to estimate the effects of GIA on vertical movements at specific GPS
points (Serpelloni et al., 2013). Modules for the computation of horizontal
displacements shall be included in future releases. Comparing column (d) with (b) and (c)
(all these runs are characterized by the intermediate-resolution configuration *R44/L128/I3*),
we note that rotational effects are important at tide gauges; however, from columns (b) and (c),
we also note that differences between the revised and the traditional rotation theories generally
do not exceed the 0.1 mm yr^{−1} level at the tide gauges considered here. Comparing outputs in column (b)
with the high-resolution run in column (a), we note maximum differences of 0.03 mm yr^{−1},
which further confirms the reliability of the test run.

Comparing the high-resolution results in Table 5 (*R100/L512/I5*, column a) with those
obtained using the ICE-6G_C (VM5a) model in the original implementation of W. R.
Peltier (see http://www.atmosp.physics.utoronto.ca/~peltier/data.php, last access: 6 June 2019),
reported in column (e), we note a fair agreement between the two model predictions.
The two sets of predictions have the same sign with the only exception of Lagos.
The differences between the values obtained with *R100/L512/I5* and the
original implementation, ICE-6G_C (VM5a), are generally close to 0.2 mm yr^{−1} or slightly larger.
The values of $\dot{\mathcal{S}}$ averaged over the tide gauges differ, but they are both <0.1 mm yr^{−1}.
More significant differences in the $\dot{\mathcal{S}}$ values are however apparent when we
compare model predictions for sites located in the polar regions beneath the former ice sheets,
which are not considered in Table 5. At these locations, the expected $\dot{\mathcal{S}}$ values
are of the order of several mm yr^{−1}, due to the large isostatic disequilibrium still associated with ice
unloading. For example, at the tide gauge of Stockholm, we obtain a rate of
−4.44 mm yr^{−1} for the high-resolution run *R100/L512/I5*, while in the original ICE-6G_C (VM5a)
implementation, the rate is −3.75 mm yr^{−1}. To better evaluate the meaning of these differences, it is worth noting
that they largely exceed the typical 2*σ*
uncertainty in the observed rates, which according to Spada and Galassi (2012) is never
larger than 0.3 mm yr^{−1} (see their Table 2).
We have also ascertained that misfits of the order of 1 mm yr^{−1} are not uncommon in
other high-latitude sites of both hemispheres. Tracing the origin of the discrepancies in the two sets of
GIA predictions (and therefore in the whole set of the GIA geodetic fingerprints considered in Fig. 8)
is not easy at this stage and would demand a detailed model intercomparison study like those performed in the GIA community
by Spada et al. (2011) and Martinec et al. (2018).
We can, however, guess that the misfit between the two sets of GIA predictions stems
from the different discretizations of the ice time histories, from the effects of mantle compressibility, and
possibly from the different rotation theories adopted.

In Fig. 9, we study the present-day rates of change of the variations
of Stokes coefficients (${\dot{\stackrel{\mathrm{\u203e}}{\mathit{\delta}c}}}_{lm},{\dot{\stackrel{\mathrm{\u203e}}{\mathit{\delta}s}}}_{lm}$) induced by GIA, computed
in the test run with *R44/L128/I3*. These quantities represent the coefficients of the expansion of the
gravity potential variation $\mathrm{\Phi}(\mathit{\gamma},t)-\mathrm{\Phi}(\mathit{\gamma},{t}_{\mathrm{0}})$ in series of spherical harmonics; hence,
they contain information upon the response
of the Earth to surface loading and to movements of the axis of rotation. In SELEN^{4}, we use a real, fully normalized
representation for the Stokes coefficients, following GRACE
conventions for spherical harmonics (see, in particular, Bettadpur, 2018
and Sect. S8.9 in SSM19). However, it is important to note that in Fig. 9,
the Stokes coefficients also include the direct effect of the Earth's rotation,
associated with the change in centrifugal potential,
on the TLNs of harmonic degree *l*=2 (i.e., they
account for the “*δ*(*t*)” term in “$\mathit{\delta}\left(t\right)+{k}_{\mathrm{2}}^{T}\left(t\right)$”; see Eq. S173); hence,
the rates we have computed are not directly
comparable with the GRACE rates. In fact, since in its orbit GRACE is not physically connected with the Earth,
it cannot be influenced by the direct rotational effect (the whole issue has been the subject of discussion a
few years ago; see Chambers et al., 2010; Peltier et al., 2012; Chambers et al., 2012). The user of SELEN^{4}, however,
can supply the program with a rotation response function (𝒢^{rot})
that does not include the direct term in order to produce GRACE-compliant
Stokes coefficients which are only indirectly affected by the Earth's rotation.

The fully normalized cosine (squares) and sine (circles) coefficients (${\dot{\stackrel{\mathrm{\u203e}}{\mathit{\delta}c}}}_{lm},{\dot{\stackrel{\mathrm{\u203e}}{\mathit{\delta}s}}}_{lm}$) are shown in Fig. 9a only for harmonics with degree
*l*≤6. The dominance of the degree 2 coefficients is apparent, which reflect the symmetries of
the $\dot{\mathcal{G}}$ fingerprint in Fig. 8d. We note that since
$\dot{\mathcal{N}}=\dot{\mathcal{G}}+\dot{c}$, where $\dot{\mathcal{N}}$ is the absolute sea-level
fingerprint in Fig. 8c
and *c* is the FC76 constant (see Eq. 22), the Stokes coefficients for $\dot{\mathcal{G}}$
and for $\dot{\mathcal{N}}$ coincide for *l*≥2. For reference, the numerical values of the degree *l*=2
coefficients obtained in the test run are
${\dot{\stackrel{\mathrm{\u203e}}{\mathit{\delta}c}}}_{\mathrm{20}}=+\mathrm{1.56}$,
${\dot{\stackrel{\mathrm{\u203e}}{\mathit{\delta}c}}}_{\mathrm{21}}=-\mathrm{0.66}$,
${\dot{\stackrel{\mathrm{\u203e}}{\mathit{\delta}s}}}_{\mathrm{21}}=+\mathrm{3.37}$,
${\dot{\stackrel{\mathrm{\u203e}}{\mathit{\delta}c}}}_{\mathrm{22}}=-\mathrm{0.34}$, and
${\dot{\stackrel{\mathrm{\u203e}}{\mathit{\delta}s}}}_{\mathrm{22}}=+\mathrm{0.05}$ in units of 10^{−11} yr^{−1}; the modulus
of all other coefficients is <1 in these units. To better visualize the decay of the Stokes coefficients with
increasing *l*, in the diagram of Fig. 9b, we show the harmonic spectrum defined in
Sect. S8.9 (see Eq. S477). By inspection of the spectrum plot, it is now apparent that the energy
contained in the degree *l*=2 harmonic component exceeds by at least 1 order of magnitude all
those with *l*≥3. After a plateau that indicates a substantial power equipartition in the range
of harmonic degrees $\mathrm{3}\le l\le \mathrm{7}$, the spectrum clearly shows a red character and decays very rapidly, closely
following a power law $\sim {l}^{-\mathrm{5}}$ (solid line). This result is consistent with those obtained
by Spada and Galassi (2015), although they have used a simplified GIA model with fixed shorelines and
the traditional rotation theory. They confirm that, for *l*≥3, the power contained in the GIA-induced regional
variations in absolute sea level is negligible when compared with the spectrum observed during the
altimetry era.

4 Conclusions

Back to toptop
We have presented an updated version of the SLE solver `SELEN`

, which was
originally introduced by Spada and Stocchi (2007) and principally meant as a tool for students.
Along with a condensed theory background and the basic features of the new program,
we have provided a step-by-step description of the outcomes of a medium-resolution test
run that requires modest computing resources. However, the run
accounts for an up-to-date description of the time history of melting since the LGM and a realistic rheological profile,
being based upon a realization of the ICE-6G_C (VM5a) model of Peltier et al. (2015).
The outputs of the test run, which cover different temporal scales, have been briefly discussed
in order to appreciate some of the possible geodynamical implications. Outputs of
a high-resolution test run have been also presented to illustrate the effects of spatial
and harmonic resolution on some GIA predictions.

With respect to the original version of the code, two major improvements have been
made in SELEN^{4}. The first is represented by an increased realism
in the description of the GIA process.
Indeed, now the program accounts for the migration of the shorelines and for the rotational feedback
on sea-level change, which enable a fully topographically and gravitationally self-consistent modelization,
in the sense defined by Peltier (1994). Furthermore, SELEN^{4} can be configured
assuming two different rotation theories or even excluding rotational effects. The second
improvement is in terms of usability, efficiency, and versatility, and covers various aspects. First, the solution of
the SLE is now performed by a single Fortran program unit, leaving to a flexible and customizable post-processor
the computation of various outputs encompassing the broad spectrum of the GIA phenomenology.
Second, on modern multi-core systems, SELEN^{4} can
take advantage of multi-threaded parallelism to speed up the most computationally intensive portions of the
code. The reader
can find details about computational aspects as execution time and code parallelism in Sect. S8.7
of SSM19. Third, the user can easily customize the time evolution of the surface load
and the rheological layering of the Earth
providing precomputed loading and tidal Love numbers.
Last, SELEN^{4} comes with a user guide and with a fully detailed theory background in a supplement,
which is particularly meant to illustrate the basic concepts of GIA to young scientists or colleagues
and to allow transparency and reproducibility.

In the framework of a recent benchmarking initiative (Martinec et al., 2018), a preliminary
version of the new program has been successfully tested against other independently developed SLE solvers, in the particular
case in which rotational effects are not taken into account and assuming simplified surface loads.
The version of SELEN^{4} that is distributed with the present work reproduces the numerical results published in
Martinec et al. (2018), if the code is configured with the parameters employed in the benchmark exercise.
After it was progressively developed in various interim versions, SELEN^{4} has now been released to the GIA and to
the global geodynamics community as an open-source tool.

Code and data availability

Back to toptop
Code and data availability.

SELEN^{4} is available from Zenodo at
https://zenodo.org/record/3520451 (https://doi.org/10.5281/zenodo.3520451; Spada and Melini, 2019)
and from the Computational
Infrastructure for Geodynamics (CIG) at http://geodynamics.org/cig/software/selen (Spada and Melini, 2015).
SELEN^{4} is released under a three-clause BSD license (for details, see
https://opensource.org/licenses/BSD-3-Clause, last access: 26 November 2019).
The ice history data for ICE-6G (VM5a) were downloaded
from http://www.atmosp.physics.utoronto.ca/~peltier/data.php (last access: 20 April 2019).
The ETOPO1 model was obtained from
https://www.ngdc.noaa.gov/mgg/global/ (last access: 26 February 2019).

Supplement

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Supplement.

The supplement related to this article is available online at: https://doi.org/10.5194/gmd-12-5055-2019-supplement.

Author contributions

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Author contributions.

GS and DM both contributed to the design and implementation of the research, to the analysis of the results, and to the writing of the manuscript. The Supplement was written by GS, with the support of DM. The code was progressively developed by GS and DM, who edited the user guide and the online version of the code.

Competing interests

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Competing interests.

The authors declare that they have no conflict of interest.

Acknowledgements

Back to toptop
Acknowledgements.

We thank Daniela Riposati from the INGV Laboratorio Grafica e Immagini for drawing the SELEN logo.
We thank Max Tegmark for having made available his pixelization routines, which have had
an essential role in the development of `SELEN`

(see https://space.mit.edu/ home/ tegmark/ isosahedron.html, last access: 26 November 2019).
We also thank Mark Wieczorek for distributing the SHTOOLS (Spherical Harmonics Tools)
to the community (https://shtools.oca.eu/shtools, last access: 26 November 2019).
Some of the figures have been drawn using the Generic Mapping Tools (GMT)
of Wessel and Smith (1998).
We are indebted to all the colleagues who
participated in the various stages of the glacial isostatic adjustment and sea-level
equation benchmark activities, namely Valentina Barletta, Paolo Gasperini, Thomas James,
Matt King, Samuel Kachuck, Volker Klemann, Björn Lund, Zdenek Martinec, Riccardo Riva, Karen Simon, Yu Sun,
Bert Vermeersen, Wouter van der Wal, and Detlef Wolf; (see Spada et al., 2011; Martinec et al., 2018).
A special acknowledgement goes to Florence Colleoni for help in the code implementation and to Francesco Mainardi
for advice on the theory of linear viscoelasticity.
We are also indebted to Michael Bevis and Erik Ivins for
encouragement and advice. We thank Glenn Milne for discussion about the meaning of the
GIA fingerprints.
We are grateful to Volker Klemann, Samuel Kachuck, and Geruo A for their constructive review and advice, and Meike Bagge and
Evan Gowan for
their very helpful comments. We have greatly benefitted from the discussion, advice, and encouragement from all the teachers and
students of the 2019 GIA Training School (26–30 August 2019, Lantmäteriet, Gävle, Sweden).
Raffaello Mascetti
has patiently revised the manuscript during various stages of its development,
also providing invaluable inspiration.

Financial support

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Financial support.

Giorgio Spada is funded by a FFABR (Finanziamento delle Attività Base di Ricerca) grant of MIUR (Ministero dell'Istruzione, dell'Università e della Ricerca) and by a research grant of DiSPeA (Dipartimento di Scienze Pure e Applicate) of the Urbino University “Carlo Bo”.

Review statement

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Review statement.

This paper was edited by Steven Phipps and reviewed by Volker Klemann, Geruo A, and Samuel Kachuck.

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Short summary

Accurate modeling of the complex physical interactions between solid Earth, oceans, and ice masses in response to deglaciation processes is of paramount importance in climate change and geodesy, since ongoing effects of the melting of Late Pleistocene ice sheets still affect present-day observations of sea-level change, uplift rates, and gravity field. In this paper, we present SELEN^{4}, an open-source code that can compute a broad range of physical predictions for a given deglaciation model.

Accurate modeling of the complex physical interactions between solid Earth, oceans, and ice...

Geoscientific Model Development

An interactive open-access journal of the European Geosciences Union