The Gravity Recovery and Climate Experiment (GRACE) mission data have an
important, if not revolutionary, impact on how scientists quantify the water
transport on the Earth's surface. The transport phenomena include land
hydrology, physical oceanography, atmospheric moisture flux, and global
cryospheric mass balance. The mass transport observed by the satellite system
also includes solid Earth motions caused by, for example, great subduction
zone earthquakes and glacial isostatic adjustment (GIA) processes. When
coupled with altimetry, these space gravimetry data provide a powerful
framework for studying climate-related changes on decadal timescales, such as
ice mass loss and sea-level rise. As the changes in the latter are
significant over the past two decades, there is a concomitant self-attraction
and loading phenomenon generating ancillary changes in gravity, sea surface,
and solid Earth deformation. These generate a finite signal in GRACE and
ocean altimetry, and it may often be desirable to isolate and remove them for
the purpose of understanding, for example, ocean circulation changes and
post-seismic viscoelastic mantle flow, or GIA, occurring beneath the
seafloor. Here we perform a systematic calculation of sea-level fingerprints
of on-land water mass changes using monthly Release-06 GRACE Level-2 Stokes
coefficients for the span April 2002 to August 2016, which result in a set of
solutions for the time-varying geoid, sea-surface height, and vertical
bedrock motion. We provide both spherical harmonic coefficients and spatial
maps of these global field variables and uncertainties therein
(

© 2019 California Institute of Technology. Government sponsorship acknowledged.

Geodesists have long understood that the ocean mean sea surface follows the
shape of the Earth's geoid

To date, space gravimetric measurements using Gravity Recovery and
Climate Experiment (GRACE) monthly gravity fields
and the subpolar ocean altimetry measurements from TOPEX/Poseidon and Jason
each have multiple geophysical signals and respective noise floors that are
generally high enough that clear detection of these contemporary land-mass-driven fingerprints in the oceans has remained elusive. However, it is
believed that these signals will eventually emerge in these data systems.
Such a belief springs, in part, from the fact that amplitudes of internal
ocean variability in intra- and interannual mass that GRACE observes are
relatively mute in comparison to on-land hydrology, two-way land-to-ocean
transport, and secular trends in land ice changes

The effects of the fingerprints are nonetheless important to disentangle from
many geophysical and ocean circulation models and dataset. New insights into
the regional and global sea-level budgets are sought through explicitly
combining ocean altimetry with the space gravimetry information, and a key
part of this combination is to account for the details of sea-level
fingerprints

This paper describes a dataset of monthly changes in relative sea level,
geoid height, and vertical bedrock motion induced by mass redistribution from
land to ocean. These are derived from the Release-06 GRACE Level-2 monthly
Stokes coefficients for the period April 2002 to August 2016. The GRACE
mission data have been instrumental to the study of the Earth's climate system

Relative sea level is defined as the height of the ocean water column bounded
by two surfaces: solid Earth surface and sea surface. Change in relative
sea level,

Mass redistributed on Earth's surface provides a direct perturbation to the
Earth's gravitational and rotational potentials, causing a corresponding
perturbation in the geoid height. Since the geoid height on a realistic Earth
does not necessarily have to coincide with the sea-surface height, we write

In geodetic applications, global field variables are typically expanded in a
spherical harmonic (SH) domain. Most of the GRACE data processing centers –
including the University of Texas Austin's Center for Space Research (CSR),
GeoForschungsZentrum Potsdam (GFZ), and Jet Propulsion Laboratory (JPL) –
provide monthly solutions for normalized SH coefficients of the gravitational
potential termed “Stokes coefficients”. Stokes coefficient anomalies – the
values that deviate from the mean (static) field – can be used to readily
retrieve changes in on-land ice and water storage or ocean bottom pressure.
The goal of this paper is to provide Stokes coefficient anomalies (i.e., SH
coefficients of

Here we briefly summarize the fundamental concept and a numerical technique
of solving the so-called “sea-level equation”. Much of the background and
supporting materials may be found, for example, in

The net change in on-land (water) mass directly affects the relative
sea level, hence conserving mass on a global scale. Such a redistribution of
mass on Earth's surface perturbs its gravitational and rotational potentials
and further redistributes the ocean mass. The net result of these
perturbations is the sea-level fingerprint: a unique spatial pattern of
relative sea level that is consistent with fundamental physical features of a
realistic Earth. For a self-gravitating elastically compressible rotating
Earth, we compute the sea-level fingerprint by satisfying the following sea-level
equation:

The barystatic term,

Changes in the Earth's surface potential,

The last term in Eq. (

To solve for the sea-level fingerprint in a conventional SH domain

We expand all of the terms appearing in Eqs. (

Once we obtain the final solution for

Here we give a brief summary of the steps undertaken to develop sea-level fingerprints and complementary data products. First, we note that the GRACE processing centers, including CSR, GFZ, and JPL, have a variety of methods employed to reduce noise, but the system has an inherent resolution limit of about 300 to 400 km in radius at the Earth's surface. Hence, the Stokes coefficients for the potential field provided by the official centers are truncated at a varying degree and order, from 60 to 96. We employ a truncation at degree and order 60, as many months may be much noisier than others.

We use GRACE Level-2 Release-06 data products provided by all three premier
(and official) data processing centers (available at

By combining GSM Stokes coefficient anomalies with GIA and low-degree
coefficients as noted above, we may derive corresponding coefficients for
land water storage anomalies,

Barystatic sea-level time series. Our estimates
of trends and seasonal amplitudes for all three data centers are compared to
JPL mascon solutions

Monthly land water storage fields,

Effects of scaling on the select spherical harmonic coefficients.

Land load function, sea-level fingerprint, and uncertainties
therein. Average rate of water equivalent height change in land water
storage

A detailed description of scaling may be found in

Comparison of data centers for select fields. JPL solutions are
subtracted from CSR and GFZ solutions for trend in land water storage
change

Based on CSR, GFZ, and JPL Stokes coefficients, we provide with this article
monthly SH coefficients of

model forcing function,

geoid height change,

vertical bedrock motion,

relative sea-level change,

We provide uncertainty associated with monthly
fields as well, both in terms of spatial maps and SH coefficients.
Quantification of the uncertainty is determined by the following recipe.
Based on the JPL Release-06 (GIA uncorrected) mascon solutions and associated
standard errors

Effects of

The utility of the data we provide
is that they may be used to rigorously remove those patterns that are
attributable to geoid height change and bedrock
motions caused by on-land mass changes from ocean altimetry, bottom-pressure,
and tide gauge studies. Such removal is essential for future studies of the
patterns of sea-level change owing to internal variability of the climate
system which drives changes in ocean density, fresh water fluxes, and
circulation

As we supply sea-level fingerprints and complementary data products with and
without rotational feedback, we owe the readers some additional words of
caution and recommendations. First, from the Eulerian equations of rotational
motion, we solve for the feedback consistently designed for periods longer
than 434 d (the period of the Chandler wobble). The rationale is that both
the solid Earth and ocean pole tide

It is also worthwhile to note that on timescales of decades the mantle
primarily behaves elastically, perhaps with the exception at places where the
tectonic history has brought heat, volatiles, and changes in mineral
structure, such as water or reduced grain size, into the region, thus
reducing the effective viscosity to values below about

We presently store data in a public repository hosted by
Harvard Dataverse (

The first set of data we supply are SH coefficients of global field
variables. The zip file “SLFsh_coefficients.zip” contains a total of 1780
data files:

The second set of data we supply are gridded maps of global field variables. We provide a total of 12 NetCDF files: four each for CSR, GFZ, and JPL. The file “SLFgrids_GFZOP_CF_WITHrotation.nc”, for example, stores solutions based on GFZ Stokes coefficients that are computed in the CF reference frame with the rotational feedback accounted for.

In this paper we describe a data product that emerges from the Release-06
GRACE Level-2 Stokes coefficients, provided by CSR, GFZ, and JPL, which
contain the basic information necessary to create monthly sea-level
fingerprints, and these are general enough that they may be employed in
reconstructions of vertical bedrock motion, perturbed relative sea surface,
and geoid height change. We provide SH coefficients of each field and
uncertainty therein, computed in both CM and CF reference frames with and
without rotational feedback included. For user convenience, we also provide
spatial maps at

A future space altimetry mission (Surface Water and Ocean Topography, or
SWOT) is aimed at providing real-time two-dimensional imaging of the
sea-surface height without the necessity of having to patch together
one-dimensional profiles

The fundamental theoretical concept of the so-called sea-level fingerprint is
summarized in Sect. 2. Here we provide explicit mathematical expressions
for all of the terms appearing in Eq. (

The barystatic term is given by

Changes in gravitational potential,

Changes in rotational potential,

The ocean-averaged term in Eq. (

Using Eqs. (

Here we briefly outline the workflow of our computation.

Given the on-land change in water equivalent height

We compute SH coefficients of the “global loading function”

Once

The choice of the solid Earth model determines the load Love numbers

The choice of the reference frame origin determines the degree 1 load Love numbers. In this study, we take the values from

Rotational feedback is accounted for via Eqs. (

As for the convergence criterion, we track the relative change in L-2 norm after each recursion and call the solution converged when it is less than 0.001 % of the L-2 norm of the solution itself. This level of solution convergence is typically achieved after four to six iterations.

Unless stated otherwise, constants and parameters used in this study are taken directly from Table 1 of

Once the solution is converged for

SA and ERI conceived the research and wrote the first draft of the paper. SA formulated the sea-level solver, with the help of ERI, and led the calculations, with the help of FWL (in estimating degree 1 Stokes coefficients) and TF and LC (in uncertainty quantification). All authors contributed to the analysis of the results and to the writing and editing of the paper.

The authors declare that they have no conflict of interest.

This research was carried out at the Jet Propulsion Laboratory (JPL), California Institute of Technology, under a contract with National Aeronautics and Space Administration (NASA). Constructive comments from the reviewers greatly improved the quality of the paper.

This research was supported by the JPL Research, Technology & Development Program (grant no. 01-STCR-R.17.235.118; 2017–2019) and the NASA Sea-Level Change Science Team (grant no. 16-SLCT16-0015; 2018–2020), as well as the NASA MEaSUREs program (grant no. 17-MEASURES-0031; 2019–2023).

This paper was edited by Giuseppe M. R. Manzella and reviewed by Don Chambers, Xuebin Zhang, and Makan A. Karegar.